sl(2)sl{}\left(2\right)-subalgebras of E8E_8

E8E_8
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.

Page generated by the calculator project.

Number of sl(2) subalgebras: 69.
Let hh be in the Cartan subalgebra. Let (α1,,αn)\left({\alpha}_{1}, \dots, {\alpha}_{n}\right) be simple roots with respect to h. Then the hh-characteristic, as defined by E. Dynkin, is the nn-tuple (α1(h),,αn(h))\left({\alpha}_{1}{}\left(h\right), \dots, {\alpha}_{n}{}\left(h\right)\right).

The actual realization of h. The coordinates of hh are given with respect to the fixed original simple basis. Note that the hh-characteristic is computed using a possibly different simple basis, more precisely, with respect to any h-positive simple basis.
A regular semisimple subalgebra might contain an sl(2)sl{}\left(2\right) such that it has no centralizer in the regular semisimple subalgebra, but the regular semisimple subalgebra might fail to be minimal containing. This happens when another minimal containing regular semisimple subalgebra of equal rank nests as a root subalgebra in the containing subalgebra. See Dynkin, Semisimple Lie subalgebras of semisimple Lie algebras, remark before Theorem 10.4.
The sl(2)sl{}\left(2\right) submodules of the ambient Lie algebra are parametrized by their highest weight with respect to the Cartan element hh of sl(2)sl{}\left(2\right). In turn, the highest weight is a positive integer multiple of the fundamental highest weight ψ\psi. Vkψ{V}_{k \psi} is k+1k+1-dimensional.


Type + realization linkh-CharacteristicRealization of hsl(2)-module decomposition of the ambient Lie algebra
ψ\psi= the fundamental sl(2)sl{}\left(2\right)-weight.
Centralizer dimensionType of semisimple part of centralizer, if knownThe square of the length of the weight dual to h.Dynkin index Minimal containing regular semisimple SAsContaining regular semisimple SAs in which the sl(2) has no centralizer
A11240A^{1240}_1(2, 2, 2, 2, 2, 2, 2, 2)(92, 136, 182, 270, 220, 168, 114, 58)V58ψ+V46ψ+V38ψ+V34ψ+V26ψ+V22ψ+V14ψ+V2ψV_{58\psi}+V_{46\psi}+V_{38\psi}+V_{34\psi}+V_{26\psi}+V_{22\psi}+V_{14\psi}+V_{2\psi}
0 0024801240E8E^{1}_8E8E^{1}_8
A1760A^{760}_1(2, 2, 2, 0, 2, 2, 2, 2)(72, 106, 142, 210, 172, 132, 90, 46)V46ψ+V38ψ+V34ψ+V28ψ+V26ψ+V22ψ+V18ψ+V14ψ+V10ψ+V2ψV_{46\psi}+V_{38\psi}+V_{34\psi}+V_{28\psi}+V_{26\psi}+V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{10\psi}+V_{2\psi}
0 001520760E8E^{1}_8E8E^{1}_8
A1520A^{520}_1(2, 2, 2, 0, 2, 0, 2, 2)(60, 88, 118, 174, 142, 108, 74, 38)V38ψ+V34ψ+V28ψ+V26ψ+2V22ψ+V18ψ+V16ψ+V14ψ+V10ψ+V6ψ+V2ψV_{38\psi}+V_{34\psi}+V_{28\psi}+V_{26\psi}+2V_{22\psi}+V_{18\psi}+V_{16\psi}+V_{14\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}
0 001040520E8E^{1}_8E8E^{1}_8
A1400A^{400}_1(2, 0, 0, 2, 0, 2, 2, 2)(52, 76, 102, 152, 124, 96, 66, 34)V34ψ+V28ψ+2V26ψ+V22ψ+2V18ψ+V16ψ+V14ψ+2V10ψ+V8ψ+2V2ψV_{34\psi}+V_{28\psi}+2V_{26\psi}+V_{22\psi}+2V_{18\psi}+V_{16\psi}+V_{14\psi}+2V_{10\psi}+V_{8\psi}+2V_{2\psi}
0 00800400E7+A1E^{1}_7+A^{1}_1E7+A1E^{1}_7+A^{1}_1, E8E^{1}_8
A1399A^{399}_1(2, 1, 1, 0, 1, 2, 2, 2)(52, 76, 102, 151, 124, 96, 66, 34)V34ψ+2V27ψ+V26ψ+V22ψ+V18ψ+2V17ψ+V14ψ+V10ψ+2V9ψ+V2ψ+3V0V_{34\psi}+2V_{27\psi}+V_{26\psi}+V_{22\psi}+V_{18\psi}+2V_{17\psi}+V_{14\psi}+V_{10\psi}+2V_{9\psi}+V_{2\psi}+3V_{0}
3 A11A^{1}_1798399E7E^{1}_7E7E^{1}_7
A1280A^{280}_1(2, 0, 0, 2, 0, 2, 0, 2)(44, 64, 86, 128, 104, 80, 54, 28)V28ψ+V26ψ+2V22ψ+2V18ψ+V16ψ+3V14ψ+2V10ψ+V8ψ+V6ψ+V4ψ+V2ψV_{28\psi}+V_{26\psi}+2V_{22\psi}+2V_{18\psi}+V_{16\psi}+3V_{14\psi}+2V_{10\psi}+V_{8\psi}+V_{6\psi}+V_{4\psi}+V_{2\psi}
0 00560280D8D^{1}_8D8D^{1}_8, E8E^{1}_8
A1232A^{232}_1(2, 0, 0, 2, 0, 0, 2, 2)(40, 58, 78, 116, 94, 72, 50, 26)V26ψ+2V22ψ+V20ψ+V18ψ+2V16ψ+2V14ψ+V12ψ+3V10ψ+2V6ψ+V4ψ+2V2ψV_{26\psi}+2V_{22\psi}+V_{20\psi}+V_{18\psi}+2V_{16\psi}+2V_{14\psi}+V_{12\psi}+3V_{10\psi}+2V_{6\psi}+V_{4\psi}+2V_{2\psi}
0 00464232E7+A1E^{1}_7+A^{1}_1E7+A1E^{1}_7+A^{1}_1, E8E^{1}_8
A1231A^{231}_1(2, 1, 1, 0, 1, 0, 2, 2)(40, 58, 78, 115, 94, 72, 50, 26)V26ψ+V22ψ+2V21ψ+V18ψ+V16ψ+2V15ψ+V14ψ+2V11ψ+2V10ψ+V6ψ+2V5ψ+V2ψ+3V0V_{26\psi}+V_{22\psi}+2V_{21\psi}+V_{18\psi}+V_{16\psi}+2V_{15\psi}+V_{14\psi}+2V_{11\psi}+2V_{10\psi}+V_{6\psi}+2V_{5\psi}+V_{2\psi}+3V_{0}
3 A11A^{1}_1462231E7E^{1}_7E7E^{1}_7
A1184A^{184}_1(2, 0, 0, 2, 0, 0, 2, 0)(36, 52, 70, 104, 84, 64, 44, 22)2V22ψ+V20ψ+V18ψ+V16ψ+3V14ψ+2V12ψ+4V10ψ+V8ψ+V6ψ+V4ψ+3V2ψ2V_{22\psi}+V_{20\psi}+V_{18\psi}+V_{16\psi}+3V_{14\psi}+2V_{12\psi}+4V_{10\psi}+V_{8\psi}+V_{6\psi}+V_{4\psi}+3V_{2\psi}
0 00368184D8D^{1}_8E8E^{1}_8, D8D^{1}_8
A1182A^{182}_1(2, 1, 1, 0, 1, 1, 0, 1)(36, 52, 70, 103, 84, 64, 43, 22)V22ψ+2V21ψ+V18ψ+2V15ψ+V14ψ+3V12ψ+2V11ψ+V10ψ+2V9ψ+V6ψ+2V3ψ+V2ψ+3V0V_{22\psi}+2V_{21\psi}+V_{18\psi}+2V_{15\psi}+V_{14\psi}+3V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+V_{6\psi}+2V_{3\psi}+V_{2\psi}+3V_{0}
3 A12A^{2}_1364182D7D^{1}_7D7D^{1}_7
A1160A^{160}_1(0, 0, 0, 2, 0, 0, 2, 2)(32, 48, 64, 96, 78, 60, 42, 22)V22ψ+2V18ψ+3V16ψ+3V14ψ+3V10ψ+3V8ψ+2V6ψ+V4ψ+4V2ψV_{22\psi}+2V_{18\psi}+3V_{16\psi}+3V_{14\psi}+3V_{10\psi}+3V_{8\psi}+2V_{6\psi}+V_{4\psi}+4V_{2\psi}
0 00320160E6+A2E^{1}_6+A^{1}_2, E7+A1E^{1}_7+A^{1}_1E6+A2E^{1}_6+A^{1}_2, E8E^{1}_8, E7+A1E^{1}_7+A^{1}_1
A1159A^{159}_1(0, 1, 1, 0, 1, 0, 2, 2)(32, 48, 64, 95, 78, 60, 42, 22)V22ψ+V18ψ+2V17ψ+V16ψ+2V15ψ+2V14ψ+2V10ψ+2V9ψ+V8ψ+2V7ψ+V6ψ+2V3ψ+2V2ψ+3V0V_{22\psi}+V_{18\psi}+2V_{17\psi}+V_{16\psi}+2V_{15\psi}+2V_{14\psi}+2V_{10\psi}+2V_{9\psi}+V_{8\psi}+2V_{7\psi}+V_{6\psi}+2V_{3\psi}+2V_{2\psi}+3V_{0}
3 A11A^{1}_1318159E7E^{1}_7E7E^{1}_7
A1157A^{157}_1(1, 0, 0, 1, 0, 1, 2, 2)(32, 47, 63, 94, 77, 60, 42, 22)V22ψ+2V17ψ+3V16ψ+2V15ψ+V14ψ+V10ψ+2V9ψ+3V8ψ+2V7ψ+2V2ψ+4Vψ+3V0V_{22\psi}+2V_{17\psi}+3V_{16\psi}+2V_{15\psi}+V_{14\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+2V_{7\psi}+2V_{2\psi}+4V_{\psi}+3V_{0}
3 A13A^{3}_1314157E6+A1E^{1}_6+A^{1}_1E6+A1E^{1}_6+A^{1}_1
A1156A^{156}_1(2, 0, 0, 0, 0, 2, 2, 2)(32, 46, 62, 92, 76, 60, 42, 22)V22ψ+7V16ψ+V14ψ+V10ψ+7V8ψ+V2ψ+14V0V_{22\psi}+7V_{16\psi}+V_{14\psi}+V_{10\psi}+7V_{8\psi}+V_{2\psi}+14V_{0}
14 G21G^{1}_2312156E6E^{1}_6E6E^{1}_6
A1120A^{120}_1(0, 0, 0, 2, 0, 0, 2, 0)(28, 42, 56, 84, 68, 52, 36, 18)2V18ψ+V16ψ+3V14ψ+3V12ψ+3V10ψ+3V8ψ+5V6ψ+V4ψ+3V2ψ2V_{18\psi}+V_{16\psi}+3V_{14\psi}+3V_{12\psi}+3V_{10\psi}+3V_{8\psi}+5V_{6\psi}+V_{4\psi}+3V_{2\psi}
0 00240120A8A^{1}_8, D8D^{1}_8A8A^{1}_8, E8E^{1}_8, D8D^{1}_8
A1112A^{112}_1(2, 0, 0, 0, 2, 0, 0, 2)(28, 40, 54, 80, 66, 50, 34, 18)V18ψ+2V16ψ+3V14ψ+V12ψ+6V10ψ+3V8ψ+3V6ψ+2V4ψ+4V2ψ+V0V_{18\psi}+2V_{16\psi}+3V_{14\psi}+V_{12\psi}+6V_{10\psi}+3V_{8\psi}+3V_{6\psi}+2V_{4\psi}+4V_{2\psi}+V_{0}
1 00224112D6+2A1D^{1}_6+2A^{1}_1, D7D^{1}_7D6+2A1D^{1}_6+2A^{1}_1, E7+A1E^{1}_7+A^{1}_1, D7D^{1}_7
A1111A^{111}_1(2, 0, 0, 1, 0, 1, 0, 2)(28, 40, 54, 80, 65, 50, 34, 18)V18ψ+V16ψ+2V15ψ+2V14ψ+2V11ψ+3V10ψ+4V9ψ+V8ψ+2V6ψ+2V5ψ+V4ψ+2V2ψ+2Vψ+3V0V_{18\psi}+V_{16\psi}+2V_{15\psi}+2V_{14\psi}+2V_{11\psi}+3V_{10\psi}+4V_{9\psi}+V_{8\psi}+2V_{6\psi}+2V_{5\psi}+V_{4\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}
3 A11A^{1}_1222111D6+A1D^{1}_6+A^{1}_1D6+A1D^{1}_6+A^{1}_1, E7E^{1}_7
A1110A^{110}_1(2, 1, 1, 0, 0, 0, 1, 2)(28, 40, 54, 79, 64, 49, 34, 18)V18ψ+4V15ψ+V14ψ+6V10ψ+4V9ψ+V6ψ+4V5ψ+V2ψ+10V0V_{18\psi}+4V_{15\psi}+V_{14\psi}+6V_{10\psi}+4V_{9\psi}+V_{6\psi}+4V_{5\psi}+V_{2\psi}+10V_{0}
10 B21B^{1}_2220110D6D^{1}_6D6D^{1}_6
A188A^{88}_1(0, 0, 0, 2, 0, 0, 0, 2)(24, 36, 48, 72, 58, 44, 30, 16)V16ψ+3V14ψ+2V12ψ+6V10ψ+3V8ψ+5V6ψ+4V4ψ+4V2ψV_{16\psi}+3V_{14\psi}+2V_{12\psi}+6V_{10\psi}+3V_{8\psi}+5V_{6\psi}+4V_{4\psi}+4V_{2\psi}
0 0017688D8D^{1}_8, E6+A2E^{1}_6+A^{1}_2E8E^{1}_8, D8D^{1}_8, E6+A2E^{1}_6+A^{1}_2
A185A^{85}_1(1, 0, 0, 1, 0, 1, 0, 2)(24, 35, 47, 70, 57, 44, 30, 16)V16ψ+V14ψ+2V13ψ+2V12ψ+2V11ψ+2V10ψ+2V9ψ+3V8ψ+2V7ψ+V6ψ+2V5ψ+3V4ψ+2V3ψ+2V2ψ+2Vψ+V0V_{16\psi}+V_{14\psi}+2V_{13\psi}+2V_{12\psi}+2V_{11\psi}+2V_{10\psi}+2V_{9\psi}+3V_{8\psi}+2V_{7\psi}+V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+V_{0}
1 0017085A7+A1A^{1}_7+A^{1}_1, E6+A1E^{1}_6+A^{1}_1A7+A1A^{1}_7+A^{1}_1, E6+A1E^{1}_6+A^{1}_1
A184A^{84}_1(2, 0, 0, 0, 0, 2, 0, 2)(24, 34, 46, 68, 56, 44, 30, 16)V16ψ+V14ψ+6V12ψ+2V10ψ+7V8ψ+V6ψ+7V4ψ+V2ψ+8V0V_{16\psi}+V_{14\psi}+6V_{12\psi}+2V_{10\psi}+7V_{8\psi}+V_{6\psi}+7V_{4\psi}+V_{2\psi}+8V_{0}
8 A21A^{1}_216884A7A^{1}_7, E6E^{1}_6A7A^{1}_7, E6E^{1}_6
A184A^{84}_1(1, 0, 0, 1, 0, 1, 1, 0)(24, 35, 47, 70, 57, 44, 30, 15)2V15ψ+V14ψ+3V12ψ+2V11ψ+V10ψ+2V9ψ+3V8ψ+4V7ψ+V6ψ+2V5ψ+3V4ψ+2V3ψ+V2ψ+3V02V_{15\psi}+V_{14\psi}+3V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+V_{2\psi}+3V_{0}
3 A14A^{4}_116884A7A^{1}_7A7A^{1}_7
A170A^{70}_1(1, 0, 0, 1, 0, 1, 0, 1)(22, 32, 43, 64, 52, 40, 27, 14)V14ψ+2V13ψ+V12ψ+2V11ψ+2V10ψ+2V9ψ+3V8ψ+4V7ψ+3V6ψ+2V5ψ+3V4ψ+2V3ψ+2V2ψ+2Vψ+V0V_{14\psi}+2V_{13\psi}+V_{12\psi}+2V_{11\psi}+2V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+3V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+V_{0}
1 0014070D5+A3D^{1}_5+A^{1}_3, D7D^{1}_7D5+A3D^{1}_5+A^{1}_3, D7D^{1}_7
A164A^{64}_1(0, 0, 0, 0, 2, 0, 0, 2)(20, 30, 40, 60, 50, 38, 26, 14)V14ψ+2V12ψ+7V10ψ+5V8ψ+5V6ψ+5V4ψ+8V2ψ+V0V_{14\psi}+2V_{12\psi}+7V_{10\psi}+5V_{8\psi}+5V_{6\psi}+5V_{4\psi}+8V_{2\psi}+V_{0}
1 0012864D5+A2D^{1}_5+A^{1}_2, D6+2A1D^{1}_6+2A^{1}_1D5+A2D^{1}_5+A^{1}_2, E7+A1E^{1}_7+A^{1}_1, D6+2A1D^{1}_6+2A^{1}_1
A163A^{63}_1(0, 0, 0, 1, 0, 1, 0, 2)(20, 30, 40, 60, 49, 38, 26, 14)V14ψ+V12ψ+2V11ψ+4V10ψ+4V9ψ+2V8ψ+2V7ψ+3V6ψ+2V5ψ+2V4ψ+4V3ψ+4V2ψ+2Vψ+3V0V_{14\psi}+V_{12\psi}+2V_{11\psi}+4V_{10\psi}+4V_{9\psi}+2V_{8\psi}+2V_{7\psi}+3V_{6\psi}+2V_{5\psi}+2V_{4\psi}+4V_{3\psi}+4V_{2\psi}+2V_{\psi}+3V_{0}
3 A11A^{1}_112663D6+A1D^{1}_6+A^{1}_1E7E^{1}_7, D6+A1D^{1}_6+A^{1}_1
A162A^{62}_1(0, 1, 1, 0, 0, 0, 1, 2)(20, 30, 40, 59, 48, 37, 26, 14)V14ψ+4V11ψ+2V10ψ+4V9ψ+5V8ψ+2V6ψ+4V5ψ+4V3ψ+6V2ψ+6V0V_{14\psi}+4V_{11\psi}+2V_{10\psi}+4V_{9\psi}+5V_{8\psi}+2V_{6\psi}+4V_{5\psi}+4V_{3\psi}+6V_{2\psi}+6V_{0}
6 2A112A^{1}_112462D5+2A1D^{1}_5+2A^{1}_1, D6D^{1}_6D5+2A1D^{1}_5+2A^{1}_1, D6D^{1}_6
A161A^{61}_1(1, 0, 0, 0, 1, 0, 1, 2)(20, 29, 39, 58, 48, 37, 26, 14)V14ψ+2V11ψ+5V10ψ+4V9ψ+3V8ψ+2V7ψ+V6ψ+2V5ψ+4V4ψ+2V3ψ+2V2ψ+6Vψ+6V0V_{14\psi}+2V_{11\psi}+5V_{10\psi}+4V_{9\psi}+3V_{8\psi}+2V_{7\psi}+V_{6\psi}+2V_{5\psi}+4V_{4\psi}+2V_{3\psi}+2V_{2\psi}+6V_{\psi}+6V_{0}
6 A12+A11A^{2}_1+A^{1}_112261D5+A1D^{1}_5+A^{1}_1D5+A1D^{1}_5+A^{1}_1
A160A^{60}_1(2, 0, 0, 0, 0, 0, 2, 2)(20, 28, 38, 56, 46, 36, 26, 14)V14ψ+9V10ψ+7V8ψ+V6ψ+8V4ψ+V2ψ+21V0V_{14\psi}+9V_{10\psi}+7V_{8\psi}+V_{6\psi}+8V_{4\psi}+V_{2\psi}+21V_{0}
21 B31B^{1}_312060D5D^{1}_5D5D^{1}_5
A157A^{57}_1(1, 0, 0, 1, 0, 1, 0, 0)(20, 29, 39, 58, 47, 36, 24, 12)3V12ψ+2V11ψ+V10ψ+2V9ψ+3V8ψ+4V7ψ+5V6ψ+4V5ψ+3V4ψ+2V3ψ+2V2ψ+2Vψ+3V03V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+5V_{6\psi}+4V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}
3 A17A^{7}_111457A6+A1A^{1}_6+A^{1}_1A6+A1A^{1}_6+A^{1}_1
A156A^{56}_1(2, 0, 0, 0, 0, 2, 0, 0)(20, 28, 38, 56, 46, 36, 24, 12)3V12ψ+5V10ψ+3V8ψ+13V6ψ+3V4ψ+5V2ψ+6V03V_{12\psi}+5V_{10\psi}+3V_{8\psi}+13V_{6\psi}+3V_{4\psi}+5V_{2\psi}+6V_{0}
6 A17+A11A^{7}_1+A^{1}_1112562D42D^{1}_4, A6A^{1}_62D42D^{1}_4, A6A^{1}_6
A140A^{40}_1(0, 0, 0, 0, 2, 0, 0, 0)(16, 24, 32, 48, 40, 30, 20, 10)4V10ψ+6V8ψ+10V6ψ+10V4ψ+10V2ψ4V_{10\psi}+6V_{8\psi}+10V_{6\psi}+10V_{4\psi}+10V_{2\psi}
0 008040A5+A2+A1A^{1}_5+A^{1}_2+A^{1}_1, 2A42A^{1}_4, D6+2A1D^{1}_6+2A^{1}_1, D5+A3D^{1}_5+A^{1}_3, 2D42D^{1}_4A5+A2+A1A^{1}_5+A^{1}_2+A^{1}_1, 2A42A^{1}_4, E8E^{1}_8, D8D^{1}_8, E7+A1E^{1}_7+A^{1}_1, E6+A2E^{1}_6+A^{1}_2, D6+2A1D^{1}_6+2A^{1}_1, D5+A3D^{1}_5+A^{1}_3, 2D42D^{1}_4
A139A^{39}_1(0, 0, 0, 1, 0, 1, 0, 0)(16, 24, 32, 48, 39, 30, 20, 10)3V10ψ+2V9ψ+3V8ψ+4V7ψ+5V6ψ+6V5ψ+4V4ψ+6V3ψ+6V2ψ+3V03V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+5V_{6\psi}+6V_{5\psi}+4V_{4\psi}+6V_{3\psi}+6V_{2\psi}+3V_{0}
3 A11A^{1}_17839A5+A2A^{1}_5+A^{1}_2, D6+A1D^{1}_6+A^{1}_1A5+A2A^{1}_5+A^{1}_2, E7E^{1}_7, D6+A1D^{1}_6+A^{1}_1
A138A^{38}_1(0, 1, 1, 0, 0, 0, 1, 0)(16, 24, 32, 47, 38, 29, 20, 10)2V10ψ+4V9ψ+V8ψ+4V7ψ+7V6ψ+4V5ψ+5V4ψ+8V3ψ+3V2ψ+6V02V_{10\psi}+4V_{9\psi}+V_{8\psi}+4V_{7\psi}+7V_{6\psi}+4V_{5\psi}+5V_{4\psi}+8V_{3\psi}+3V_{2\psi}+6V_{0}
6 2A112A^{1}_17638D4+A3D^{1}_4+A^{1}_3, D6D^{1}_6D4+A3D^{1}_4+A^{1}_3, D6D^{1}_6
A137A^{37}_1(1, 0, 0, 0, 1, 0, 1, 0)(16, 23, 31, 46, 38, 29, 20, 10)2V10ψ+2V9ψ+4V8ψ+4V7ψ+4V6ψ+6V5ψ+7V4ψ+4V3ψ+4V2ψ+4Vψ+3V02V_{10\psi}+2V_{9\psi}+4V_{8\psi}+4V_{7\psi}+4V_{6\psi}+6V_{5\psi}+7V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{\psi}+3V_{0}
3 A13A^{3}_17437A5+2A1A^{1}_5+2A^{1}_1A5+2A1A^{1}_5+2A^{1}_1, E6+A1E^{1}_6+A^{1}_1
A136A^{36}_1(2, 0, 0, 0, 0, 0, 2, 0)(16, 22, 30, 44, 36, 28, 20, 10)2V10ψ+8V8ψ+8V6ψ+15V4ψ+3V2ψ+14V02V_{10\psi}+8V_{8\psi}+8V_{6\psi}+15V_{4\psi}+3V_{2\psi}+14V_{0}
14 G21G^{1}_27236A5+A1A^{1}_5+A^{1}_1A5+A1A^{1}_5+A^{1}_1, E6E^{1}_6
A136A^{36}_1(1, 0, 0, 1, 0, 0, 0, 1)(16, 23, 31, 46, 37, 28, 19, 10)V10ψ+4V9ψ+3V8ψ+2V7ψ+5V6ψ+8V5ψ+7V4ψ+4V3ψ+2V2ψ+4Vψ+6V0V_{10\psi}+4V_{9\psi}+3V_{8\psi}+2V_{7\psi}+5V_{6\psi}+8V_{5\psi}+7V_{4\psi}+4V_{3\psi}+2V_{2\psi}+4V_{\psi}+6V_{0}
6 A13+A11A^{3}_1+A^{1}_17236A5+A1A^{1}_5+A^{1}_1A5+A1A^{1}_5+A^{1}_1
A135A^{35}_1(2, 0, 0, 0, 0, 1, 0, 1)(16, 22, 30, 44, 36, 28, 19, 10)V10ψ+2V9ψ+7V8ψ+V6ψ+14V5ψ+7V4ψ+2V3ψ+V2ψ+17V0V_{10\psi}+2V_{9\psi}+7V_{8\psi}+V_{6\psi}+14V_{5\psi}+7V_{4\psi}+2V_{3\psi}+V_{2\psi}+17V_{0}
17 G21+A11G^{1}_2+A^{1}_17035A5A^{1}_5A5A^{1}_5
A134A^{34}_1(0, 0, 1, 0, 0, 1, 0, 1)(15, 22, 30, 44, 36, 28, 19, 10)V10ψ+2V9ψ+3V8ψ+6V7ψ+4V6ψ+6V5ψ+6V4ψ+4V3ψ+7V2ψ+4Vψ+3V0V_{10\psi}+2V_{9\psi}+3V_{8\psi}+6V_{7\psi}+4V_{6\psi}+6V_{5\psi}+6V_{4\psi}+4V_{3\psi}+7V_{2\psi}+4V_{\psi}+3V_{0}
3 A16A^{6}_16834D5+A2D^{1}_5+A^{1}_2D5+A2D^{1}_5+A^{1}_2
A132A^{32}_1(0, 2, 0, 0, 0, 0, 0, 2)(14, 22, 28, 42, 34, 26, 18, 10)V10ψ+6V8ψ+14V6ψ+7V4ψ+14V2ψ+8V0V_{10\psi}+6V_{8\psi}+14V_{6\psi}+7V_{4\psi}+14V_{2\psi}+8V_{0}
8 A22A^{2}_26432D4+4A1D^{1}_4+4A^{1}_1, D4+A2D^{1}_4+A^{1}_2, D5+2A1D^{1}_5+2A^{1}_1D4+4A1D^{1}_4+4A^{1}_1, D4+A2D^{1}_4+A^{1}_2, D5+2A1D^{1}_5+2A^{1}_1
A131A^{31}_1(0, 0, 0, 1, 0, 0, 0, 2)(14, 21, 28, 42, 34, 26, 18, 10)V10ψ+3V8ψ+6V7ψ+8V6ψ+6V5ψ+3V4ψ+2V3ψ+7V2ψ+10Vψ+6V0V_{10\psi}+3V_{8\psi}+6V_{7\psi}+8V_{6\psi}+6V_{5\psi}+3V_{4\psi}+2V_{3\psi}+7V_{2\psi}+10V_{\psi}+6V_{0}
6 A18+A11A^{8}_1+A^{1}_16231D4+3A1D^{1}_4+3A^{1}_1, D5+A1D^{1}_5+A^{1}_1D4+3A1D^{1}_4+3A^{1}_1, D5+A1D^{1}_5+A^{1}_1
A130A^{30}_1(1, 0, 0, 0, 0, 1, 0, 2)(14, 20, 27, 40, 33, 26, 18, 10)V10ψ+V8ψ+8V7ψ+8V6ψ+8V5ψ+V4ψ+8V2ψ+8Vψ+15V0V_{10\psi}+V_{8\psi}+8V_{7\psi}+8V_{6\psi}+8V_{5\psi}+V_{4\psi}+8V_{2\psi}+8V_{\psi}+15V_{0}
15 A31A^{1}_36030D4+2A1D^{1}_4+2A^{1}_1, D5D^{1}_5D4+2A1D^{1}_4+2A^{1}_1, D5D^{1}_5
A130A^{30}_1(0, 0, 0, 1, 0, 0, 1, 0)(14, 21, 28, 42, 34, 26, 18, 9)2V9ψ+3V8ψ+4V7ψ+6V6ψ+6V5ψ+6V4ψ+8V3ψ+6V2ψ+4Vψ+3V02V_{9\psi}+3V_{8\psi}+4V_{7\psi}+6V_{6\psi}+6V_{5\psi}+6V_{4\psi}+8V_{3\psi}+6V_{2\psi}+4V_{\psi}+3V_{0}
3 A110A^{10}_16030A4+A3A^{1}_4+A^{1}_3A4+A3A^{1}_4+A^{1}_3
A129A^{29}_1(0, 1, 0, 0, 0, 0, 1, 2)(13, 20, 26, 39, 32, 25, 18, 10)V10ψ+6V7ψ+14V6ψ+6V5ψ+2V2ψ+14Vψ+21V0V_{10\psi}+6V_{7\psi}+14V_{6\psi}+6V_{5\psi}+2V_{2\psi}+14V_{\psi}+21V_{0}
21 C31C^{1}_35829D4+A1D^{1}_4+A^{1}_1D4+A1D^{1}_4+A^{1}_1
A128A^{28}_1(0, 0, 0, 0, 0, 0, 2, 2)(12, 18, 24, 36, 30, 24, 18, 10)V10ψ+26V6ψ+V2ψ+52V0V_{10\psi}+26V_{6\psi}+V_{2\psi}+52V_{0}
52 F41F^{1}_45628D4D^{1}_4D4D^{1}_4
A125A^{25}_1(0, 0, 1, 0, 0, 1, 0, 0)(13, 19, 26, 38, 31, 24, 16, 8)3V8ψ+4V7ψ+5V6ψ+6V5ψ+10V4ψ+8V3ψ+7V2ψ+6Vψ+3V03V_{8\psi}+4V_{7\psi}+5V_{6\psi}+6V_{5\psi}+10V_{4\psi}+8V_{3\psi}+7V_{2\psi}+6V_{\psi}+3V_{0}
3 A115A^{15}_15025A4+A2+A1A^{1}_4+A^{1}_2+A^{1}_1A4+A2+A1A^{1}_4+A^{1}_2+A^{1}_1
A124A^{24}_1(0, 0, 0, 0, 0, 2, 0, 0)(12, 18, 24, 36, 30, 24, 16, 8)3V8ψ+13V6ψ+14V4ψ+18V2ψ+6V03V_{8\psi}+13V_{6\psi}+14V_{4\psi}+18V_{2\psi}+6V_{0}
6 A115+A11A^{15}_1+A^{1}_14824A4+A2A^{1}_4+A^{1}_2, 2D42D^{1}_4A4+A2A^{1}_4+A^{1}_2, 2D42D^{1}_4
A122A^{22}_1(0, 0, 0, 1, 0, 0, 0, 1)(12, 18, 24, 36, 29, 22, 15, 8)V8ψ+4V7ψ+5V6ψ+8V5ψ+9V4ψ+8V3ψ+9V2ψ+8Vψ+4V0V_{8\psi}+4V_{7\psi}+5V_{6\psi}+8V_{5\psi}+9V_{4\psi}+8V_{3\psi}+9V_{2\psi}+8V_{\psi}+4V_{0}
4 A12A^{2}_144222A3+2A12A^{1}_3+2A^{1}_1, A4+2A1A^{1}_4+2A^{1}_1, D4+A3D^{1}_4+A^{1}_32A3+2A12A^{1}_3+2A^{1}_1, A4+2A1A^{1}_4+2A^{1}_1, D4+A3D^{1}_4+A^{1}_3
A121A^{21}_1(1, 0, 0, 0, 0, 1, 0, 1)(12, 17, 23, 34, 28, 22, 15, 8)V8ψ+2V7ψ+7V6ψ+8V5ψ+9V4ψ+8V3ψ+8V2ψ+8Vψ+9V0V_{8\psi}+2V_{7\psi}+7V_{6\psi}+8V_{5\psi}+9V_{4\psi}+8V_{3\psi}+8V_{2\psi}+8V_{\psi}+9V_{0}
9 A21A^{1}_242212A3+A12A^{1}_3+A^{1}_1, A4+A1A^{1}_4+A^{1}_12A3+A12A^{1}_3+A^{1}_1, A4+A1A^{1}_4+A^{1}_1
A120A^{20}_1(2, 0, 0, 0, 0, 0, 0, 2)(12, 16, 22, 32, 26, 20, 14, 8)V8ψ+11V6ψ+21V4ψ+11V2ψ+24V0V_{8\psi}+11V_{6\psi}+21V_{4\psi}+11V_{2\psi}+24V_{0}
24 A41A^{1}_440202A32A^{1}_3, A4A^{1}_42A32A^{1}_3, A4A^{1}_4
A120A^{20}_1(1, 0, 0, 0, 1, 0, 0, 0)(12, 17, 23, 34, 28, 21, 14, 7)4V7ψ+6V6ψ+4V5ψ+10V4ψ+16V3ψ+6V2ψ+4Vψ+10V04V_{7\psi}+6V_{6\psi}+4V_{5\psi}+10V_{4\psi}+16V_{3\psi}+6V_{2\psi}+4V_{\psi}+10V_{0}
10 B22B^{2}_240202A32A^{1}_32A32A^{1}_3
A116A^{16}_1(0, 2, 0, 0, 0, 0, 0, 0)(10, 16, 20, 30, 24, 18, 12, 6)8V6ψ+20V4ψ+28V2ψ+8V08V_{6\psi}+20V_{4\psi}+28V_{2\psi}+8V_{0}
8 A26A^{6}_23216D4+A2D^{1}_4+A^{1}_2, D4+4A1D^{1}_4+4A^{1}_1, 4A24A^{1}_2, A3+A2+2A1A^{1}_3+A^{1}_2+2A^{1}_1D4+A2D^{1}_4+A^{1}_2, D4+4A1D^{1}_4+4A^{1}_1, 4A24A^{1}_2, A3+A2+2A1A^{1}_3+A^{1}_2+2A^{1}_1
A115A^{15}_1(0, 0, 0, 1, 0, 0, 0, 0)(10, 15, 20, 30, 24, 18, 12, 6)5V6ψ+6V5ψ+10V4ψ+14V3ψ+15V2ψ+10Vψ+6V05V_{6\psi}+6V_{5\psi}+10V_{4\psi}+14V_{3\psi}+15V_{2\psi}+10V_{\psi}+6V_{0}
6 A124+A11A^{24}_1+A^{1}_13015A3+A2+A1A^{1}_3+A^{1}_2+A^{1}_1, D4+3A1D^{1}_4+3A^{1}_1A3+A2+A1A^{1}_3+A^{1}_2+A^{1}_1, D4+3A1D^{1}_4+3A^{1}_1
A114A^{14}_1(1, 0, 0, 0, 0, 1, 0, 0)(10, 14, 19, 28, 23, 18, 12, 6)3V6ψ+8V5ψ+8V4ψ+16V3ψ+16V2ψ+8Vψ+11V03V_{6\psi}+8V_{5\psi}+8V_{4\psi}+16V_{3\psi}+16V_{2\psi}+8V_{\psi}+11V_{0}
11 B21B^{1}_22814A3+4A1A^{1}_3+4A^{1}_1, A3+A2A^{1}_3+A^{1}_2, D4+2A1D^{1}_4+2A^{1}_1A3+4A1A^{1}_3+4A^{1}_1, A3+A2A^{1}_3+A^{1}_2, D4+2A1D^{1}_4+2A^{1}_1
A113A^{13}_1(0, 1, 0, 0, 0, 0, 1, 0)(9, 14, 18, 27, 22, 17, 12, 6)2V6ψ+6V5ψ+13V4ψ+12V3ψ+16V2ψ+14Vψ+9V02V_{6\psi}+6V_{5\psi}+13V_{4\psi}+12V_{3\psi}+16V_{2\psi}+14V_{\psi}+9V_{0}
9 3A113A^{1}_126133A2+A13A^{1}_2+A^{1}_1, A3+3A1A^{1}_3+3A^{1}_1, D4+A1D^{1}_4+A^{1}_13A2+A13A^{1}_2+A^{1}_1, A3+3A1A^{1}_3+3A^{1}_1, D4+A1D^{1}_4+A^{1}_1
A112A^{12}_1(0, 0, 1, 0, 0, 0, 0, 1)(9, 13, 18, 26, 21, 16, 11, 6)V6ψ+6V5ψ+11V4ψ+16V3ψ+15V2ψ+14Vψ+13V0V_{6\psi}+6V_{5\psi}+11V_{4\psi}+16V_{3\psi}+15V_{2\psi}+14V_{\psi}+13V_{0}
13 B21+A12B^{1}_2+A^{2}_12412A3+2A1A^{1}_3+2A^{1}_1A3+2A1A^{1}_3+2A^{1}_1
A112A^{12}_1(0, 0, 0, 0, 0, 0, 2, 0)(8, 12, 16, 24, 20, 16, 12, 6)2V6ψ+25V4ψ+27V2ψ+28V02V_{6\psi}+25V_{4\psi}+27V_{2\psi}+28V_{0}
28 D41D^{1}_424123A23A^{1}_2, A3+2A1A^{1}_3+2A^{1}_1, D4D^{1}_43A23A^{1}_2, A3+2A1A^{1}_3+2A^{1}_1, D4D^{1}_4
A111A^{11}_1(0, 0, 0, 0, 0, 1, 0, 1)(8, 12, 16, 24, 20, 16, 11, 6)V6ψ+2V5ψ+15V4ψ+18V3ψ+10V2ψ+14Vψ+24V0V_{6\psi}+2V_{5\psi}+15V_{4\psi}+18V_{3\psi}+10V_{2\psi}+14V_{\psi}+24V_{0}
24 B31+A11B^{1}_3+A^{1}_12211A3+A1A^{1}_3+A^{1}_1A3+A1A^{1}_3+A^{1}_1
A110A^{10}_1(1, 0, 0, 0, 0, 0, 0, 2)(8, 11, 15, 22, 18, 14, 10, 6)V6ψ+11V4ψ+32V3ψ+V2ψ+55V0V_{6\psi}+11V_{4\psi}+32V_{3\psi}+V_{2\psi}+55V_{0}
55 B51B^{1}_52010A3A^{1}_3A3A^{1}_3
A110A^{10}_1(0, 0, 0, 0, 1, 0, 0, 0)(8, 12, 16, 24, 20, 15, 10, 5)4V5ψ+10V4ψ+16V3ψ+20V2ψ+20Vψ+10V04V_{5\psi}+10V_{4\psi}+16V_{3\psi}+20V_{2\psi}+20V_{\psi}+10V_{0}
10 B23B^{3}_220102A2+2A12A^{1}_2+2A^{1}_12A2+2A12A^{1}_2+2A^{1}_1
A19A^{9}_1(1, 0, 0, 0, 0, 0, 1, 0)(8, 11, 15, 22, 18, 14, 10, 5)2V5ψ+10V4ψ+16V3ψ+23V2ψ+18Vψ+17V02V_{5\psi}+10V_{4\psi}+16V_{3\psi}+23V_{2\psi}+18V_{\psi}+17V_{0}
17 G21+A13G^{1}_2+A^{3}_11892A2+A12A^{1}_2+A^{1}_12A2+A12A^{1}_2+A^{1}_1
A18A^{8}_1(2, 0, 0, 0, 0, 0, 0, 0)(8, 10, 14, 20, 16, 12, 8, 4)14V4ψ+50V2ψ+28V014V_{4\psi}+50V_{2\psi}+28V_{0}
28 2G212G^{1}_21682A22A^{1}_2, 8A18A^{1}_1, A2+4A1A^{1}_2+4A^{1}_12A22A^{1}_2, 8A18A^{1}_1, A2+4A1A^{1}_2+4A^{1}_1
A17A^{7}_1(0, 0, 1, 0, 0, 0, 0, 0)(7, 10, 14, 20, 16, 12, 8, 4)7V4ψ+14V3ψ+28V2ψ+28Vψ+17V07V_{4\psi}+14V_{3\psi}+28V_{2\psi}+28V_{\psi}+17V_{0}
17 G22+A11G^{2}_2+A^{1}_11477A17A^{1}_1, A2+3A1A^{1}_2+3A^{1}_17A17A^{1}_1, A2+3A1A^{1}_2+3A^{1}_1
A16A^{6}_1(0, 0, 0, 0, 0, 1, 0, 0)(6, 9, 12, 18, 15, 12, 8, 4)3V4ψ+16V3ψ+27V2ψ+32Vψ+24V03V_{4\psi}+16V_{3\psi}+27V_{2\psi}+32V_{\psi}+24V_{0}
24 B31+A16B^{1}_3+A^{6}_1126A2+2A1A^{1}_2+2A^{1}_1, 6A16A^{1}_1A2+2A1A^{1}_2+2A^{1}_1, 6A16A^{1}_1
A15A^{5}_1(1, 0, 0, 0, 0, 0, 0, 1)(6, 8, 11, 16, 13, 10, 7, 4)V4ψ+12V3ψ+32V2ψ+32Vψ+35V0V_{4\psi}+12V_{3\psi}+32V_{2\psi}+32V_{\psi}+35V_{0}
35 A51A^{1}_51055A15A^{1}_1, A2+A1A^{1}_2+A^{1}_15A15A^{1}_1, A2+A1A^{1}_2+A^{1}_1
A14A^{4}_1(0, 0, 0, 0, 0, 0, 0, 2)(4, 6, 8, 12, 10, 8, 6, 4)V4ψ+55V2ψ+78V0V_{4\psi}+55V_{2\psi}+78V_{0}
78 E61E^{1}_6844A14A^{1}_1, A2A^{1}_24A14A^{1}_1, A2A^{1}_2
A14A^{4}_1(0, 1, 0, 0, 0, 0, 0, 0)(5, 8, 10, 15, 12, 9, 6, 3)8V3ψ+28V2ψ+48Vψ+36V08V_{3\psi}+28V_{2\psi}+48V_{\psi}+36V_{0}
36 C41C^{1}_4844A14A^{1}_14A14A^{1}_1
A13A^{3}_1(0, 0, 0, 0, 0, 0, 1, 0)(4, 6, 8, 12, 10, 8, 6, 3)2V3ψ+27V2ψ+52Vψ+55V02V_{3\psi}+27V_{2\psi}+52V_{\psi}+55V_{0}
55 F41+A11F^{1}_4+A^{1}_1633A13A^{1}_13A13A^{1}_1
A12A^{2}_1(1, 0, 0, 0, 0, 0, 0, 0)(4, 5, 7, 10, 8, 6, 4, 2)14V2ψ+64Vψ+78V014V_{2\psi}+64V_{\psi}+78V_{0}
78 B61B^{1}_6422A12A^{1}_12A12A^{1}_1
A1A_1(0, 0, 0, 0, 0, 0, 0, 1)(2, 3, 4, 6, 5, 4, 3, 2)V2ψ+56Vψ+133V0V_{2\psi}+56V_{\psi}+133V_{0}
133 E71E^{1}_721A1A^{1}_1A1A^{1}_1



Length longest root ambient algebra squared/4= 1/2

Given a root subsystem PP, and a root sub-subsystem P0{P}_{0}, in (10.2) of Semisimple subalgebras of semisimple Lie algebras, E. Dynkin defines a numerical constant e(P,P0)e{}\left(P, {P}_{0}\right) (which we call Dynkin epsilon).
In Theorem 10.3, Dynkin proves that if an sl(2)sl{}\left(2\right) is an SS-subalgebra in the root subalgebra generated by PP, such that it has characteristic 2 for all simple roots of PP lying in P0{P}_{0}, then e{}\left(P, {P}_{0}\right)=0e{}\left(P, {P}_{0}\right)=0.
H that wasn't realized on the first attempt but was ultimately realized: (52, 76, 102, 152, 124, 96, 66, 34), Type: A1400{A^{400}}_{1}.
It turns out that in the current case of Cartan element h = (52, 76, 102, 152, 124, 96, 66, 34) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (44, 64, 86, 128, 104, 80, 54, 28), Type: A1280{A^{280}}_{1}.
It turns out that in the current case of Cartan element h = (44, 64, 86, 128, 104, 80, 54, 28) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (40, 58, 78, 116, 94, 72, 50, 26), Type: A1232{A^{232}}_{1}.
It turns out that in the current case of Cartan element h = (40, 58, 78, 116, 94, 72, 50, 26) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (36, 52, 70, 104, 84, 64, 44, 22), Type: A1184{A^{184}}_{1}.
It turns out that in the current case of Cartan element h = (36, 52, 70, 104, 84, 64, 44, 22) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (32, 48, 64, 96, 78, 60, 42, 22), Type: A1160{A^{160}}_{1}.
It turns out that in the current case of Cartan element h = (32, 48, 64, 96, 78, 60, 42, 22) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (28, 42, 56, 84, 68, 52, 36, 18), Type: A1120{A^{120}}_{1}.
It turns out that in the current case of Cartan element h = (28, 42, 56, 84, 68, 52, 36, 18) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (28, 40, 54, 80, 66, 50, 34, 18), Type: A1112{A^{112}}_{1}.
It turns out that in the current case of Cartan element h = (28, 40, 54, 80, 66, 50, 34, 18) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (28, 40, 54, 80, 65, 50, 34, 18), Type: A1111{A^{111}}_{1}.
It turns out that in the current case of Cartan element h = (28, 40, 54, 80, 65, 50, 34, 18) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (24, 36, 48, 72, 58, 44, 30, 16), Type: A188{A^{88}}_{1}.
It turns out that in the current case of Cartan element h = (24, 36, 48, 72, 58, 44, 30, 16) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (24, 35, 47, 70, 57, 44, 30, 16), Type: A185{A^{85}}_{1}.
It turns out that in the current case of Cartan element h = (24, 35, 47, 70, 57, 44, 30, 16) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (24, 34, 46, 68, 56, 44, 30, 16), Type: A184{A^{84}}_{1}.
It turns out that in the current case of Cartan element h = (24, 34, 46, 68, 56, 44, 30, 16) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (22, 32, 43, 64, 52, 40, 27, 14), Type: A170{A^{70}}_{1}.
It turns out that in the current case of Cartan element h = (22, 32, 43, 64, 52, 40, 27, 14) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (20, 30, 40, 60, 50, 38, 26, 14), Type: A164{A^{64}}_{1}.
It turns out that in the current case of Cartan element h = (20, 30, 40, 60, 50, 38, 26, 14) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (20, 30, 40, 60, 49, 38, 26, 14), Type: A163{A^{63}}_{1}.
It turns out that in the current case of Cartan element h = (20, 30, 40, 60, 49, 38, 26, 14) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (20, 30, 40, 59, 48, 37, 26, 14), Type: A162{A^{62}}_{1}.
It turns out that in the current case of Cartan element h = (20, 30, 40, 59, 48, 37, 26, 14) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (16, 24, 32, 48, 40, 30, 20, 10), Type: A140{A^{40}}_{1}.
It turns out that in the current case of Cartan element h = (16, 24, 32, 48, 40, 30, 20, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (16, 24, 32, 48, 39, 30, 20, 10), Type: A139{A^{39}}_{1}.
It turns out that in the current case of Cartan element h = (16, 24, 32, 48, 39, 30, 20, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (16, 24, 32, 47, 38, 29, 20, 10), Type: A138{A^{38}}_{1}.
It turns out that in the current case of Cartan element h = (16, 24, 32, 47, 38, 29, 20, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (16, 23, 31, 46, 38, 29, 20, 10), Type: A137{A^{37}}_{1}.
It turns out that in the current case of Cartan element h = (16, 23, 31, 46, 38, 29, 20, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (16, 22, 30, 44, 36, 28, 20, 10), Type: A136{A^{36}}_{1}.
It turns out that in the current case of Cartan element h = (16, 22, 30, 44, 36, 28, 20, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (14, 22, 28, 42, 34, 26, 18, 10), Type: A132{A^{32}}_{1}.
It turns out that in the current case of Cartan element h = (14, 22, 28, 42, 34, 26, 18, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (14, 21, 28, 42, 34, 26, 18, 10), Type: A131{A^{31}}_{1}.
It turns out that in the current case of Cartan element h = (14, 21, 28, 42, 34, 26, 18, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (14, 20, 27, 40, 33, 26, 18, 10), Type: A130{A^{30}}_{1}.
It turns out that in the current case of Cartan element h = (14, 20, 27, 40, 33, 26, 18, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (12, 18, 24, 36, 30, 24, 16, 8), Type: A124{A^{24}}_{1}.
It turns out that in the current case of Cartan element h = (12, 18, 24, 36, 30, 24, 16, 8) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (12, 18, 24, 36, 29, 22, 15, 8), Type: A122{A^{22}}_{1}.
It turns out that in the current case of Cartan element h = (12, 18, 24, 36, 29, 22, 15, 8) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 20, 30, 24, 18, 12, 6), Type: A116{A^{16}}_{1}.
It turns out that in the current case of Cartan element h = (10, 16, 20, 30, 24, 18, 12, 6) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (10, 15, 20, 30, 24, 18, 12, 6), Type: A115{A^{15}}_{1}.
It turns out that in the current case of Cartan element h = (10, 15, 20, 30, 24, 18, 12, 6) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (10, 14, 19, 28, 23, 18, 12, 6), Type: A114{A^{14}}_{1}.
It turns out that in the current case of Cartan element h = (10, 14, 19, 28, 23, 18, 12, 6) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (9, 14, 18, 27, 22, 17, 12, 6), Type: A113{A^{13}}_{1}.
It turns out that in the current case of Cartan element h = (9, 14, 18, 27, 22, 17, 12, 6) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (8, 12, 16, 24, 20, 16, 12, 6), Type: A112{A^{12}}_{1}.
It turns out that in the current case of Cartan element h = (8, 12, 16, 24, 20, 16, 12, 6) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (8, 10, 14, 20, 16, 12, 8, 4), Type: A18{A^{8}}_{1}.
It turns out that in the current case of Cartan element h = (8, 10, 14, 20, 16, 12, 8, 4) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
H that wasn't realized on the first attempt but was ultimately realized: (6, 9, 12, 18, 15, 12, 8, 4), Type: A16{A^{6}}_{1}.
It turns out that in the current case of Cartan element h = (6, 9, 12, 18, 15, 12, 8, 4) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.
Extensions of the rationals used (4 total): \mathbb Q, [7]\mathbb Q[\sqrt{7}], [2]\mathbb Q[\sqrt{2}], [1, 2, 7]\mathbb Q[\sqrt{-1}, \sqrt{2}, \sqrt{7}]
A11240A^{1240}_1
h-characteristic: (2, 2, 2, 2, 2, 2, 2, 2)
Length of the weight dual to h: 2480
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: E8E^{1}_8
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V58ψ+V46ψ+V38ψ+V34ψ+V26ψ+V22ψ+V14ψ+V2ψV_{58\psi}+V_{46\psi}+V_{38\psi}+V_{34\psi}+V_{26\psi}+V_{22\psi}+V_{14\psi}+V_{2\psi}
Below is one possible realization of the sl(2) subalgebra.
h=58h8+114h7+168h6+220h5+270h4+182h3+136h2+92h1e=58g8+114g7+168g6+220g5+270g4+182g3+136g2+92g1f=g1+g2+g3+g4+g5+g6+g7+g8\begin{array}{rcl}h&=&58h_{8}+114h_{7}+168h_{6}+220h_{5}+270h_{4}+182h_{3}+136h_{2}+92h_{1}\\ e&=&58g_{8}+114g_{7}+168g_{6}+220g_{5}+270g_{4}+182g_{3}+136g_{2}+92g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}+g_{-7}+g_{-8}\end{array}
Lie brackets of the above elements.
h=58h8+114h7+168h6+220h5+270h4+182h3+136h2+92h1e=58g8+114g7+168g6+220g5+270g4+182g3+136g2+92g1f=g1+g2+g3+g4+g5+g6+g7+g8\begin{array}{rcl}[e, f]&=&58h_{8}+114h_{7}+168h_{6}+220h_{5}+270h_{4}+182h_{3}+136h_{2}+92h_{1}\\ [h, e]&=&116g_{8}+228g_{7}+336g_{6}+440g_{5}+540g_{4}+364g_{3}+272g_{2}+184g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-6}-2g_{-7}-2g_{-8}\end{array}
Centralizer type: 00
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Unknown elements.
h=58h8+114h7+168h6+220h5+270h4+182h3+136h2+92h1e=x8g8+x7g7+x6g6+x5g5+x4g4+x3g3+x2g2+x1g1e=x9g1+x10g2+x11g3+x12g4+x13g5+x14g6+x15g7+x16g8\begin{array}{rcl}h&=&58h_{8}+114h_{7}+168h_{6}+220h_{5}+270h_{4}+182h_{3}+136h_{2}+92h_{1}\\ e&=&x_{8} g_{8}+x_{7} g_{7}+x_{6} g_{6}+x_{5} g_{5}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{9} g_{-1}+x_{10} g_{-2}+x_{11} g_{-3}+x_{12} g_{-4}+x_{13} g_{-5}+x_{14} g_{-6}+x_{15} g_{-7}+x_{16} g_{-8}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x8x1658)h8+(x7x15114)h7+(x6x14168)h6+(x5x13220)h5+(x4x12270)h4+(x3x11182)h3+(x2x10136)h2+(x1x992)h1[e,f] - h = \left(x_{8} x_{16} -58\right)h_{8}+\left(x_{7} x_{15} -114\right)h_{7}+\left(x_{6} x_{14} -168\right)h_{6}+\left(x_{5} x_{13} -220\right)h_{5}+\left(x_{4} x_{12} -270\right)h_{4}+\left(x_{3} x_{11} -182\right)h_{3}+\left(x_{2} x_{10} -136\right)h_{2}+\left(x_{1} x_{9} -92\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x992=0x2x10136=0x3x11182=0x4x12270=0x5x13220=0x6x14168=0x7x15114=0x8x1658=0\begin{array}{rcl}x_{1} x_{9} -92&=&0\\x_{2} x_{10} -136&=&0\\x_{3} x_{11} -182&=&0\\x_{4} x_{12} -270&=&0\\x_{5} x_{13} -220&=&0\\x_{6} x_{14} -168&=&0\\x_{7} x_{15} -114&=&0\\x_{8} x_{16} -58&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=58h8+114h7+168h6+220h5+270h4+182h3+136h2+92h1e=x8g8+x7g7+x6g6+x5g5+x4g4+x3g3+x2g2+x1g1f=g1+g2+g3+g4+g5+g6+g7+g8\begin{array}{rcl}h&=&58h_{8}+114h_{7}+168h_{6}+220h_{5}+270h_{4}+182h_{3}+136h_{2}+92h_{1}\\e&=&x_{8} g_{8}+x_{7} g_{7}+x_{6} g_{6}+x_{5} g_{5}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}+g_{-7}+g_{-8}\end{array}
Matrix form of the system we are trying to solve: (1000000001000000001000000001000000001000000001000000001000000001)[col. vect.]=(9213618227022016811458)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}92\\ 136\\ 182\\ 270\\ 220\\ 168\\ 114\\ 58\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=58h8+114h7+168h6+220h5+270h4+182h3+136h2+92h1e=x8g8+x7g7+x6g6+x5g5+x4g4+x3g3+x2g2+x1g1f=x9g1+x10g2+x11g3+x12g4+x13g5+x14g6+x15g7+x16g8\begin{array}{rcl}h&=&58h_{8}+114h_{7}+168h_{6}+220h_{5}+270h_{4}+182h_{3}+136h_{2}+92h_{1}\\ e&=&x_{8} g_{8}+x_{7} g_{7}+x_{6} g_{6}+x_{5} g_{5}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{9} g_{-1}+x_{10} g_{-2}+x_{11} g_{-3}+x_{12} g_{-4}+x_{13} g_{-5}+x_{14} g_{-6}+x_{15} g_{-7}+x_{16} g_{-8}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x992=0x2x10136=0x3x11182=0x4x12270=0x5x13220=0x6x14168=0x7x15114=0x8x1658=0\begin{array}{rcl}x_{1} x_{9} -92&=&0\\x_{2} x_{10} -136&=&0\\x_{3} x_{11} -182&=&0\\x_{4} x_{12} -270&=&0\\x_{5} x_{13} -220&=&0\\x_{6} x_{14} -168&=&0\\x_{7} x_{15} -114&=&0\\x_{8} x_{16} -58&=&0\\\end{array}

A1760A^{760}_1
h-characteristic: (2, 2, 2, 0, 2, 2, 2, 2)
Length of the weight dual to h: 1520
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: E8E^{1}_8
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V46ψ+V38ψ+V34ψ+V28ψ+V26ψ+V22ψ+V18ψ+V14ψ+V10ψ+V2ψV_{46\psi}+V_{38\psi}+V_{34\psi}+V_{28\psi}+V_{26\psi}+V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{10\psi}+V_{2\psi}
Below is one possible realization of the sl(2) subalgebra.
h=46h8+90h7+132h6+172h5+210h4+142h3+106h2+72h1e=172g12+19g1119g10+46g890g7+132g636g5+123g3+87g2+72g1f=g1+g2+g3+g6g7+g8g10+g11g12\begin{array}{rcl}h&=&46h_{8}+90h_{7}+132h_{6}+172h_{5}+210h_{4}+142h_{3}+106h_{2}+72h_{1}\\ e&=&-172g_{12}+19g_{11}-19g_{10}+46g_{8}-90g_{7}+132g_{6}-36g_{5}+123g_{3}+87g_{2}+72g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-6}-g_{-7}+g_{-8}-g_{-10}+g_{-11}-g_{-12}\end{array}
Lie brackets of the above elements.
h=46h8+90h7+132h6+172h5+210h4+142h3+106h2+72h1e=172g12+19g1119g10+46g890g7+132g636g5+123g3+87g2+72g1f=g1+g2+g3+g6g7+g8g10+g11g12\begin{array}{rcl}[e, f]&=&46h_{8}+90h_{7}+132h_{6}+172h_{5}+210h_{4}+142h_{3}+106h_{2}+72h_{1}\\ [h, e]&=&-344g_{12}+38g_{11}-38g_{10}+92g_{8}-180g_{7}+264g_{6}-72g_{5}+246g_{3}+174g_{2}+144g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-6}+2g_{-7}-2g_{-8}+2g_{-10}-2g_{-11}+2g_{-12}\end{array}
Centralizer type: 00
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Unknown elements.
h=46h8+90h7+132h6+172h5+210h4+142h3+106h2+72h1e=x4g12+x3g11+x2g10+x7g8+x6g7+x5g6+x10g5+x9g3+x8g2+x1g1e=x11g1+x18g2+x19g3+x20g5+x15g6+x16g7+x17g8+x12g10+x13g11+x14g12\begin{array}{rcl}h&=&46h_{8}+90h_{7}+132h_{6}+172h_{5}+210h_{4}+142h_{3}+106h_{2}+72h_{1}\\ e&=&x_{4} g_{12}+x_{3} g_{11}+x_{2} g_{10}+x_{7} g_{8}+x_{6} g_{7}+x_{5} g_{6}+x_{10} g_{5}+x_{9} g_{3}+x_{8} g_{2}+x_{1} g_{1}\\ f&=&x_{11} g_{-1}+x_{18} g_{-2}+x_{19} g_{-3}+x_{20} g_{-5}+x_{15} g_{-6}+x_{16} g_{-7}+x_{17} g_{-8}+x_{12} g_{-10}+x_{13} g_{-11}+x_{14} g_{-12}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x18x3x19+x4x20)g4+(x7x1746)h8+(x6x1690)h7+(x5x15132)h6+(x4x14+x10x20172)h5+(x2x12+x3x13+x4x14210)h4+(x3x13+x9x19142)h3+(x2x12+x8x18106)h2+(x1x1172)h1+(x8x12x9x13+x10x14)g4[e,f] - h = \left(-x_{2} x_{18} -x_{3} x_{19} +x_{4} x_{20} \right)g_{4}+\left(x_{7} x_{17} -46\right)h_{8}+\left(x_{6} x_{16} -90\right)h_{7}+\left(x_{5} x_{15} -132\right)h_{6}+\left(x_{4} x_{14} +x_{10} x_{20} -172\right)h_{5}+\left(x_{2} x_{12} +x_{3} x_{13} +x_{4} x_{14} -210\right)h_{4}+\left(x_{3} x_{13} +x_{9} x_{19} -142\right)h_{3}+\left(x_{2} x_{12} +x_{8} x_{18} -106\right)h_{2}+\left(x_{1} x_{11} -72\right)h_{1}+\left(-x_{8} x_{12} -x_{9} x_{13} +x_{10} x_{14} \right)g_{-4}
The polynomial system that corresponds to finding the h, e, f triple:
x1x1172=0x2x12+x8x18106=0x2x12+x3x13+x4x14210=0x2x18x3x19+x4x20=0x3x13+x9x19142=0x4x14+x10x20172=0x5x15132=0x6x1690=0x7x1746=0x8x12x9x13+x10x14=0\begin{array}{rcl}x_{1} x_{11} -72&=&0\\x_{2} x_{12} +x_{8} x_{18} -106&=&0\\x_{2} x_{12} +x_{3} x_{13} +x_{4} x_{14} -210&=&0\\-x_{2} x_{18} -x_{3} x_{19} +x_{4} x_{20} &=&0\\x_{3} x_{13} +x_{9} x_{19} -142&=&0\\x_{4} x_{14} +x_{10} x_{20} -172&=&0\\x_{5} x_{15} -132&=&0\\x_{6} x_{16} -90&=&0\\x_{7} x_{17} -46&=&0\\-x_{8} x_{12} -x_{9} x_{13} +x_{10} x_{14} &=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=46h8+90h7+132h6+172h5+210h4+142h3+106h2+72h1e=x4g12+x3g11+x2g10+x7g8+x6g7+x5g6+x10g5+x9g3+x8g2+x1g1f=g1+g2+g3+g6g7+g8g10+g11g12\begin{array}{rcl}h&=&46h_{8}+90h_{7}+132h_{6}+172h_{5}+210h_{4}+142h_{3}+106h_{2}+72h_{1}\\e&=&x_{4} g_{12}+x_{3} g_{11}+x_{2} g_{10}+x_{7} g_{8}+x_{6} g_{7}+x_{5} g_{6}+x_{10} g_{5}+x_{9} g_{3}+x_{8} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-3}+g_{-6}-g_{-7}+g_{-8}-g_{-10}+g_{-11}-g_{-12}\end{array}
Matrix form of the system we are trying to solve: (1000000000010000010001110000000110000000001000001000010000000000100000000001000000000010000000000111)[col. vect.]=(72106210014217213290460)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}72\\ 106\\ 210\\ 0\\ 142\\ 172\\ 132\\ 90\\ 46\\ 0\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=46h8+90h7+132h6+172h5+210h4+142h3+106h2+72h1e=x4g12+x3g11+x2g10+x7g8+x6g7+x5g6+x10g5+x9g3+x8g2+x1g1f=x11g1+x18g2+x19g3+x20g5+x15g6+x16g7+x17g8+x12g10+x13g11+x14g12\begin{array}{rcl}h&=&46h_{8}+90h_{7}+132h_{6}+172h_{5}+210h_{4}+142h_{3}+106h_{2}+72h_{1}\\ e&=&x_{4} g_{12}+x_{3} g_{11}+x_{2} g_{10}+x_{7} g_{8}+x_{6} g_{7}+x_{5} g_{6}+x_{10} g_{5}+x_{9} g_{3}+x_{8} g_{2}+x_{1} g_{1}\\ f&=&x_{11} g_{-1}+x_{18} g_{-2}+x_{19} g_{-3}+x_{20} g_{-5}+x_{15} g_{-6}+x_{16} g_{-7}+x_{17} g_{-8}+x_{12} g_{-10}+x_{13} g_{-11}+x_{14} g_{-12}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x1172=0x2x12+x8x18106=0x2x12+x3x13+x4x14210=0x2x18x3x19+x4x20=0x3x13+x9x19142=0x4x14+x10x20172=0x5x15132=0x6x1690=0x7x1746=0x8x12x9x13+x10x14=0\begin{array}{rcl}x_{1} x_{11} -72&=&0\\x_{2} x_{12} +x_{8} x_{18} -106&=&0\\x_{2} x_{12} +x_{3} x_{13} +x_{4} x_{14} -210&=&0\\-x_{2} x_{18} -x_{3} x_{19} +x_{4} x_{20} &=&0\\x_{3} x_{13} +x_{9} x_{19} -142&=&0\\x_{4} x_{14} +x_{10} x_{20} -172&=&0\\x_{5} x_{15} -132&=&0\\x_{6} x_{16} -90&=&0\\x_{7} x_{17} -46&=&0\\-x_{8} x_{12} -x_{9} x_{13} +x_{10} x_{14} &=&0\\\end{array}

A1520A^{520}_1
h-characteristic: (2, 2, 2, 0, 2, 0, 2, 2)
Length of the weight dual to h: 1040
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: E8E^{1}_8
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V38ψ+V34ψ+V28ψ+V26ψ+2V22ψ+V18ψ+V16ψ+V14ψ+V10ψ+V6ψ+V2ψV_{38\psi}+V_{34\psi}+V_{28\psi}+V_{26\psi}+2V_{22\psi}+V_{18\psi}+V_{16\psi}+V_{14\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}
Below is one possible realization of the sl(2) subalgebra.
h=38h8+74h7+108h6+142h5+174h4+118h3+88h2+60h1e=2g20+52g14+54g13+32g12+85g1155g1038g8+22g7+54g5+33g3+33g2+60g1f=g1+g2+g3+g5+g7g8g10+g11+g12+g13+g14g20\begin{array}{rcl}h&=&38h_{8}+74h_{7}+108h_{6}+142h_{5}+174h_{4}+118h_{3}+88h_{2}+60h_{1}\\ e&=&-2g_{20}+52g_{14}+54g_{13}+32g_{12}+85g_{11}-55g_{10}-38g_{8}+22g_{7}+54g_{5}+33g_{3}+33g_{2}+60g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-5}+g_{-7}-g_{-8}-g_{-10}+g_{-11}+g_{-12}+g_{-13}+g_{-14}-g_{-20}\end{array}
Lie brackets of the above elements.
h=38h8+74h7+108h6+142h5+174h4+118h3+88h2+60h1e=2g20+52g14+54g13+32g12+85g1155g1038g8+22g7+54g5+33g3+33g2+60g1f=g1+g2+g3+g5+g7g8g10+g11+g12+g13+g14g20\begin{array}{rcl}[e, f]&=&38h_{8}+74h_{7}+108h_{6}+142h_{5}+174h_{4}+118h_{3}+88h_{2}+60h_{1}\\ [h, e]&=&-4g_{20}+104g_{14}+108g_{13}+64g_{12}+170g_{11}-110g_{10}-76g_{8}+44g_{7}+108g_{5}+66g_{3}+66g_{2}+120g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-5}-2g_{-7}+2g_{-8}+2g_{-10}-2g_{-11}-2g_{-12}-2g_{-13}-2g_{-14}+2g_{-20}\end{array}
Centralizer type: 00
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Unknown elements.
h=38h8+74h7+108h6+142h5+174h4+118h3+88h2+60h1e=x4g20+x5g14+x9g13+x10g12+x3g11+x2g10+x6g8+x11g7+x12g5+x8g3+x7g2+x1g1e=x13g1+x19g2+x20g3+x24g5+x23g7+x18g8+x14g10+x15g11+x22g12+x21g13+x17g14+x16g20\begin{array}{rcl}h&=&38h_{8}+74h_{7}+108h_{6}+142h_{5}+174h_{4}+118h_{3}+88h_{2}+60h_{1}\\ e&=&x_{4} g_{20}+x_{5} g_{14}+x_{9} g_{13}+x_{10} g_{12}+x_{3} g_{11}+x_{2} g_{10}+x_{6} g_{8}+x_{11} g_{7}+x_{12} g_{5}+x_{8} g_{3}+x_{7} g_{2}+x_{1} g_{1}\\ f&=&x_{13} g_{-1}+x_{19} g_{-2}+x_{20} g_{-3}+x_{24} g_{-5}+x_{23} g_{-7}+x_{18} g_{-8}+x_{14} g_{-10}+x_{15} g_{-11}+x_{22} g_{-12}+x_{21} g_{-13}+x_{17} g_{-14}+x_{16} g_{-20}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x4x22+x5x23x9x24)g6+(x2x19x3x20+x4x21+x10x24)g4+(x6x1838)h8+(x5x17+x11x2374)h7+(x4x16+x5x17+x9x21108)h6+(x4x16+x9x21+x10x22+x12x24142)h5+(x2x14+x3x15+x4x16+x10x22174)h4+(x3x15+x8x20118)h3+(x2x14+x7x1988)h2+(x1x1360)h1+(x7x14x8x15+x9x16+x12x22)g4+(x10x16+x11x17x12x21)g6[e,f] - h = \left(-x_{4} x_{22} +x_{5} x_{23} -x_{9} x_{24} \right)g_{6}+\left(-x_{2} x_{19} -x_{3} x_{20} +x_{4} x_{21} +x_{10} x_{24} \right)g_{4}+\left(x_{6} x_{18} -38\right)h_{8}+\left(x_{5} x_{17} +x_{11} x_{23} -74\right)h_{7}+\left(x_{4} x_{16} +x_{5} x_{17} +x_{9} x_{21} -108\right)h_{6}+\left(x_{4} x_{16} +x_{9} x_{21} +x_{10} x_{22} +x_{12} x_{24} -142\right)h_{5}+\left(x_{2} x_{14} +x_{3} x_{15} +x_{4} x_{16} +x_{10} x_{22} -174\right)h_{4}+\left(x_{3} x_{15} +x_{8} x_{20} -118\right)h_{3}+\left(x_{2} x_{14} +x_{7} x_{19} -88\right)h_{2}+\left(x_{1} x_{13} -60\right)h_{1}+\left(-x_{7} x_{14} -x_{8} x_{15} +x_{9} x_{16} +x_{12} x_{22} \right)g_{-4}+\left(-x_{10} x_{16} +x_{11} x_{17} -x_{12} x_{21} \right)g_{-6}
The polynomial system that corresponds to finding the h, e, f triple:
x1x1360=0x2x14+x7x1988=0x2x14+x3x15+x4x16+x10x22174=0x2x19x3x20+x4x21+x10x24=0x3x15+x8x20118=0x4x16+x9x21+x10x22+x12x24142=0x4x16+x5x17+x9x21108=0x4x22+x5x23x9x24=0x5x17+x11x2374=0x6x1838=0x7x14x8x15+x9x16+x12x22=0x10x16+x11x17x12x21=0\begin{array}{rcl}x_{1} x_{13} -60&=&0\\x_{2} x_{14} +x_{7} x_{19} -88&=&0\\x_{2} x_{14} +x_{3} x_{15} +x_{4} x_{16} +x_{10} x_{22} -174&=&0\\-x_{2} x_{19} -x_{3} x_{20} +x_{4} x_{21} +x_{10} x_{24} &=&0\\x_{3} x_{15} +x_{8} x_{20} -118&=&0\\x_{4} x_{16} +x_{9} x_{21} +x_{10} x_{22} +x_{12} x_{24} -142&=&0\\x_{4} x_{16} +x_{5} x_{17} +x_{9} x_{21} -108&=&0\\-x_{4} x_{22} +x_{5} x_{23} -x_{9} x_{24} &=&0\\x_{5} x_{17} +x_{11} x_{23} -74&=&0\\x_{6} x_{18} -38&=&0\\-x_{7} x_{14} -x_{8} x_{15} +x_{9} x_{16} +x_{12} x_{22} &=&0\\-x_{10} x_{16} +x_{11} x_{17} -x_{12} x_{21} &=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=38h8+74h7+108h6+142h5+174h4+118h3+88h2+60h1e=x4g20+x5g14+x9g13+x10g12+x3g11+x2g10+x6g8+x11g7+x12g5+x8g3+x7g2+x1g1f=g1+g2+g3+g5+g7g8g10+g11+g12+g13+g14g20\begin{array}{rcl}h&=&38h_{8}+74h_{7}+108h_{6}+142h_{5}+174h_{4}+118h_{3}+88h_{2}+60h_{1}\\e&=&x_{4} g_{20}+x_{5} g_{14}+x_{9} g_{13}+x_{10} g_{12}+x_{3} g_{11}+x_{2} g_{10}+x_{6} g_{8}+x_{11} g_{7}+x_{12} g_{5}+x_{8} g_{3}+x_{7} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-3}+g_{-5}+g_{-7}-g_{-8}-g_{-10}+g_{-11}+g_{-12}+g_{-13}+g_{-14}-g_{-20}\end{array}
Matrix form of the system we are trying to solve: (100000000000010000100000011100000100011100000100001000010000000100001101000110001000000110001000000010000010000001000000000000111001000000000111)[col. vect.]=(608817401181421080743800)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & -1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1\\ 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}60\\ 88\\ 174\\ 0\\ 118\\ 142\\ 108\\ 0\\ 74\\ 38\\ 0\\ 0\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=38h8+74h7+108h6+142h5+174h4+118h3+88h2+60h1e=x4g20+x5g14+x9g13+x10g12+x3g11+x2g10+x6g8+x11g7+x12g5+x8g3+x7g2+x1g1f=x13g1+x19g2+x20g3+x24g5+x23g7+x18g8+x14g10+x15g11+x22g12+x21g13+x17g14+x16g20\begin{array}{rcl}h&=&38h_{8}+74h_{7}+108h_{6}+142h_{5}+174h_{4}+118h_{3}+88h_{2}+60h_{1}\\ e&=&x_{4} g_{20}+x_{5} g_{14}+x_{9} g_{13}+x_{10} g_{12}+x_{3} g_{11}+x_{2} g_{10}+x_{6} g_{8}+x_{11} g_{7}+x_{12} g_{5}+x_{8} g_{3}+x_{7} g_{2}+x_{1} g_{1}\\ f&=&x_{13} g_{-1}+x_{19} g_{-2}+x_{20} g_{-3}+x_{24} g_{-5}+x_{23} g_{-7}+x_{18} g_{-8}+x_{14} g_{-10}+x_{15} g_{-11}+x_{22} g_{-12}+x_{21} g_{-13}+x_{17} g_{-14}+x_{16} g_{-20}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x1360=0x2x14+x7x1988=0x2x14+x3x15+x4x16+x10x22174=0x2x19x3x20+x4x21+x10x24=0x3x15+x8x20118=0x4x16+x9x21+x10x22+x12x24142=0x4x16+x5x17+x9x21108=0x4x22+x5x23x9x24=0x5x17+x11x2374=0x6x1838=0x7x14x8x15+x9x16+x12x22=0x10x16+x11x17x12x21=0\begin{array}{rcl}x_{1} x_{13} -60&=&0\\x_{2} x_{14} +x_{7} x_{19} -88&=&0\\x_{2} x_{14} +x_{3} x_{15} +x_{4} x_{16} +x_{10} x_{22} -174&=&0\\-x_{2} x_{19} -x_{3} x_{20} +x_{4} x_{21} +x_{10} x_{24} &=&0\\x_{3} x_{15} +x_{8} x_{20} -118&=&0\\x_{4} x_{16} +x_{9} x_{21} +x_{10} x_{22} +x_{12} x_{24} -142&=&0\\x_{4} x_{16} +x_{5} x_{17} +x_{9} x_{21} -108&=&0\\-x_{4} x_{22} +x_{5} x_{23} -x_{9} x_{24} &=&0\\x_{5} x_{17} +x_{11} x_{23} -74&=&0\\x_{6} x_{18} -38&=&0\\-x_{7} x_{14} -x_{8} x_{15} +x_{9} x_{16} +x_{12} x_{22} &=&0\\-x_{10} x_{16} +x_{11} x_{17} -x_{12} x_{21} &=&0\\\end{array}

A1400A^{400}_1
h-characteristic: (2, 0, 0, 2, 0, 2, 2, 2)
Length of the weight dual to h: 800
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: E7+A1E^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 2: E8E^{1}_8
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V34ψ+V28ψ+2V26ψ+V22ψ+2V18ψ+V16ψ+V14ψ+2V10ψ+V8ψ+2V2ψV_{34\psi}+V_{28\psi}+2V_{26\psi}+V_{22\psi}+2V_{18\psi}+V_{16\psi}+V_{14\psi}+2V_{10\psi}+V_{8\psi}+2V_{2\psi}
Below is one possible realization of the sl(2) subalgebra.
h=34h8+66h7+96h6+124h5+152h4+102h3+76h2+52h1e=27g19+g18+75g17+96g13+34g8+66g7+49g4+52g1f=g1+g4+g7+g8+g13+g17+g18+g19\begin{array}{rcl}h&=&34h_{8}+66h_{7}+96h_{6}+124h_{5}+152h_{4}+102h_{3}+76h_{2}+52h_{1}\\ e&=&27g_{19}+g_{18}+75g_{17}+96g_{13}+34g_{8}+66g_{7}+49g_{4}+52g_{1}\\ f&=&g_{-1}+g_{-4}+g_{-7}+g_{-8}+g_{-13}+g_{-17}+g_{-18}+g_{-19}\end{array}
Lie brackets of the above elements.
h=34h8+66h7+96h6+124h5+152h4+102h3+76h2+52h1e=27g19+g18+75g17+96g13+34g8+66g7+49g4+52g1f=g1+g4+g7+g8+g13+g17+g18+g19\begin{array}{rcl}[e, f]&=&34h_{8}+66h_{7}+96h_{6}+124h_{5}+152h_{4}+102h_{3}+76h_{2}+52h_{1}\\ [h, e]&=&54g_{19}+2g_{18}+150g_{17}+192g_{13}+68g_{8}+132g_{7}+98g_{4}+104g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-4}-2g_{-7}-2g_{-8}-2g_{-13}-2g_{-17}-2g_{-18}-2g_{-19}\end{array}
Centralizer type: 00
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Unknown elements.
h=34h8+66h7+96h6+124h5+152h4+102h3+76h2+52h1e=x7g19+x8g18+x5g17+x4g13+x1g8+x3g7+x2g4+x6g1e=x14g1+x10g4+x11g7+x9g8+x12g13+x13g17+x16g18+x15g19\begin{array}{rcl}h&=&34h_{8}+66h_{7}+96h_{6}+124h_{5}+152h_{4}+102h_{3}+76h_{2}+52h_{1}\\ e&=&x_{7} g_{19}+x_{8} g_{18}+x_{5} g_{17}+x_{4} g_{13}+x_{1} g_{8}+x_{3} g_{7}+x_{2} g_{4}+x_{6} g_{1}\\ f&=&x_{14} g_{-1}+x_{10} g_{-4}+x_{11} g_{-7}+x_{9} g_{-8}+x_{12} g_{-13}+x_{13} g_{-17}+x_{16} g_{-18}+x_{15} g_{-19}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x934)h8+(x3x1166)h7+(x4x1296)h6+(x4x12+x7x15+x8x16124)h5+(x2x10+x5x13+x7x15+x8x16152)h4+(x5x13+x7x15102)h3+(x5x13+x8x1676)h2+(x6x1452)h1[e,f] - h = \left(x_{1} x_{9} -34\right)h_{8}+\left(x_{3} x_{11} -66\right)h_{7}+\left(x_{4} x_{12} -96\right)h_{6}+\left(x_{4} x_{12} +x_{7} x_{15} +x_{8} x_{16} -124\right)h_{5}+\left(x_{2} x_{10} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -152\right)h_{4}+\left(x_{5} x_{13} +x_{7} x_{15} -102\right)h_{3}+\left(x_{5} x_{13} +x_{8} x_{16} -76\right)h_{2}+\left(x_{6} x_{14} -52\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x934=0x2x10+x5x13+x7x15+x8x16152=0x3x1166=0x4x12+x7x15+x8x16124=0x4x1296=0x5x13+x8x1676=0x5x13+x7x15102=0x6x1452=0\begin{array}{rcl}x_{1} x_{9} -34&=&0\\x_{2} x_{10} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -152&=&0\\x_{3} x_{11} -66&=&0\\x_{4} x_{12} +x_{7} x_{15} +x_{8} x_{16} -124&=&0\\x_{4} x_{12} -96&=&0\\x_{5} x_{13} +x_{8} x_{16} -76&=&0\\x_{5} x_{13} +x_{7} x_{15} -102&=&0\\x_{6} x_{14} -52&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=34h8+66h7+96h6+124h5+152h4+102h3+76h2+52h1e=x7g19+x8g18+x5g17+x4g13+x1g8+x3g7+x2g4+x6g1f=g1+g4+g7+g8+g13+g17+g18+g19\begin{array}{rcl}h&=&34h_{8}+66h_{7}+96h_{6}+124h_{5}+152h_{4}+102h_{3}+76h_{2}+52h_{1}\\e&=&x_{7} g_{19}+x_{8} g_{18}+x_{5} g_{17}+x_{4} g_{13}+x_{1} g_{8}+x_{3} g_{7}+x_{2} g_{4}+x_{6} g_{1}\\f&=&g_{-1}+g_{-4}+g_{-7}+g_{-8}+g_{-13}+g_{-17}+g_{-18}+g_{-19}\end{array}
Matrix form of the system we are trying to solve: (1000000001001011001000000001001100010000000010010000101000000100)[col. vect.]=(3415266124967610252)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}34\\ 152\\ 66\\ 124\\ 96\\ 76\\ 102\\ 52\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=34h8+66h7+96h6+124h5+152h4+102h3+76h2+52h1e=x7g19+x8g18+x5g17+x4g13+x1g8+x3g7+x2g4+x6g1f=x14g1+x10g4+x11g7+x9g8+x12g13+x13g17+x16g18+x15g19\begin{array}{rcl}h&=&34h_{8}+66h_{7}+96h_{6}+124h_{5}+152h_{4}+102h_{3}+76h_{2}+52h_{1}\\ e&=&x_{7} g_{19}+x_{8} g_{18}+x_{5} g_{17}+x_{4} g_{13}+x_{1} g_{8}+x_{3} g_{7}+x_{2} g_{4}+x_{6} g_{1}\\ f&=&x_{14} g_{-1}+x_{10} g_{-4}+x_{11} g_{-7}+x_{9} g_{-8}+x_{12} g_{-13}+x_{13} g_{-17}+x_{16} g_{-18}+x_{15} g_{-19}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x934=0x2x10+x5x13+x7x15+x8x16152=0x3x1166=0x4x12+x7x15+x8x16124=0x4x1296=0x5x13+x8x1676=0x5x13+x7x15102=0x6x1452=0\begin{array}{rcl}x_{1} x_{9} -34&=&0\\x_{2} x_{10} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -152&=&0\\x_{3} x_{11} -66&=&0\\x_{4} x_{12} +x_{7} x_{15} +x_{8} x_{16} -124&=&0\\x_{4} x_{12} -96&=&0\\x_{5} x_{13} +x_{8} x_{16} -76&=&0\\x_{5} x_{13} +x_{7} x_{15} -102&=&0\\x_{6} x_{14} -52&=&0\\\end{array}

A1399A^{399}_1
h-characteristic: (2, 1, 1, 0, 1, 2, 2, 2)
Length of the weight dual to h: 798
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: E7E^{1}_7
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V34ψ+2V27ψ+V26ψ+V22ψ+V18ψ+2V17ψ+V14ψ+V10ψ+2V9ψ+V2ψ+3V0V_{34\psi}+2V_{27\psi}+V_{26\psi}+V_{22\psi}+V_{18\psi}+2V_{17\psi}+V_{14\psi}+V_{10\psi}+2V_{9\psi}+V_{2\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=34h8+66h7+96h6+124h5+151h4+102h3+76h2+52h1e=75g19+49g18+27g17+34g8+66g7+96g6+52g1f=g1+g6+g7+g8+g17+g18+g19\begin{array}{rcl}h&=&34h_{8}+66h_{7}+96h_{6}+124h_{5}+151h_{4}+102h_{3}+76h_{2}+52h_{1}\\ e&=&75g_{19}+49g_{18}+27g_{17}+34g_{8}+66g_{7}+96g_{6}+52g_{1}\\ f&=&g_{-1}+g_{-6}+g_{-7}+g_{-8}+g_{-17}+g_{-18}+g_{-19}\end{array}
Lie brackets of the above elements.
h=34h8+66h7+96h6+124h5+151h4+102h3+76h2+52h1e=75g19+49g18+27g17+34g8+66g7+96g6+52g1f=g1+g6+g7+g8+g17+g18+g19\begin{array}{rcl}[e, f]&=&34h_{8}+66h_{7}+96h_{6}+124h_{5}+151h_{4}+102h_{3}+76h_{2}+52h_{1}\\ [h, e]&=&150g_{19}+98g_{18}+54g_{17}+68g_{8}+132g_{7}+192g_{6}+104g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-6}-2g_{-7}-2g_{-8}-2g_{-17}-2g_{-18}-2g_{-19}\end{array}
Centralizer type: A1A_1
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Unknown elements.
h=34h8+66h7+96h6+124h5+151h4+102h3+76h2+52h1e=x5g19+x2g18+x7g17+x1g8+x3g7+x4g6+x6g1e=x13g1+x11g6+x10g7+x8g8+x14g17+x9g18+x12g19\begin{array}{rcl}h&=&34h_{8}+66h_{7}+96h_{6}+124h_{5}+151h_{4}+102h_{3}+76h_{2}+52h_{1}\\ e&=&x_{5} g_{19}+x_{2} g_{18}+x_{7} g_{17}+x_{1} g_{8}+x_{3} g_{7}+x_{4} g_{6}+x_{6} g_{1}\\ f&=&x_{13} g_{-1}+x_{11} g_{-6}+x_{10} g_{-7}+x_{8} g_{-8}+x_{14} g_{-17}+x_{9} g_{-18}+x_{12} g_{-19}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x834)h8+(x3x1066)h7+(x4x1196)h6+(x2x9+x5x12124)h5+(x2x9+x5x12+x7x14151)h4+(x5x12+x7x14102)h3+(x2x9+x7x1476)h2+(x6x1352)h1[e,f] - h = \left(x_{1} x_{8} -34\right)h_{8}+\left(x_{3} x_{10} -66\right)h_{7}+\left(x_{4} x_{11} -96\right)h_{6}+\left(x_{2} x_{9} +x_{5} x_{12} -124\right)h_{5}+\left(x_{2} x_{9} +x_{5} x_{12} +x_{7} x_{14} -151\right)h_{4}+\left(x_{5} x_{12} +x_{7} x_{14} -102\right)h_{3}+\left(x_{2} x_{9} +x_{7} x_{14} -76\right)h_{2}+\left(x_{6} x_{13} -52\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x834=0x2x9+x7x1476=0x2x9+x5x12+x7x14151=0x2x9+x5x12124=0x3x1066=0x4x1196=0x5x12+x7x14102=0x6x1352=0\begin{array}{rcl}x_{1} x_{8} -34&=&0\\x_{2} x_{9} +x_{7} x_{14} -76&=&0\\x_{2} x_{9} +x_{5} x_{12} +x_{7} x_{14} -151&=&0\\x_{2} x_{9} +x_{5} x_{12} -124&=&0\\x_{3} x_{10} -66&=&0\\x_{4} x_{11} -96&=&0\\x_{5} x_{12} +x_{7} x_{14} -102&=&0\\x_{6} x_{13} -52&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=34h8+66h7+96h6+124h5+151h4+102h3+76h2+52h1e=x5g19+x2g18+x7g17+x1g8+x3g7+x4g6+x6g1f=g1+g6+g7+g8+g17+g18+g19\begin{array}{rcl}h&=&34h_{8}+66h_{7}+96h_{6}+124h_{5}+151h_{4}+102h_{3}+76h_{2}+52h_{1}\\e&=&x_{5} g_{19}+x_{2} g_{18}+x_{7} g_{17}+x_{1} g_{8}+x_{3} g_{7}+x_{4} g_{6}+x_{6} g_{1}\\f&=&g_{-1}+g_{-6}+g_{-7}+g_{-8}+g_{-17}+g_{-18}+g_{-19}\end{array}
Matrix form of the system we are trying to solve: (10000000100001010010101001000010000000100000001010000010)[col. vect.]=(3476151124669610252)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}34\\ 76\\ 151\\ 124\\ 66\\ 96\\ 102\\ 52\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=34h8+66h7+96h6+124h5+151h4+102h3+76h2+52h1e=x5g19+x2g18+x7g17+x1g8+x3g7+x4g6+x6g1f=x13g1+x11g6+x10g7+x8g8+x14g17+x9g18+x12g19\begin{array}{rcl}h&=&34h_{8}+66h_{7}+96h_{6}+124h_{5}+151h_{4}+102h_{3}+76h_{2}+52h_{1}\\ e&=&x_{5} g_{19}+x_{2} g_{18}+x_{7} g_{17}+x_{1} g_{8}+x_{3} g_{7}+x_{4} g_{6}+x_{6} g_{1}\\ f&=&x_{13} g_{-1}+x_{11} g_{-6}+x_{10} g_{-7}+x_{8} g_{-8}+x_{14} g_{-17}+x_{9} g_{-18}+x_{12} g_{-19}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x834=0x2x9+x7x1476=0x2x9+x5x12+x7x14151=0x2x9+x5x12124=0x3x1066=0x4x1196=0x5x12+x7x14102=0x6x1352=0\begin{array}{rcl}x_{1} x_{8} -34&=&0\\x_{2} x_{9} +x_{7} x_{14} -76&=&0\\x_{2} x_{9} +x_{5} x_{12} +x_{7} x_{14} -151&=&0\\x_{2} x_{9} +x_{5} x_{12} -124&=&0\\x_{3} x_{10} -66&=&0\\x_{4} x_{11} -96&=&0\\x_{5} x_{12} +x_{7} x_{14} -102&=&0\\x_{6} x_{13} -52&=&0\\\end{array}

A1280A^{280}_1
h-characteristic: (2, 0, 0, 2, 0, 2, 0, 2)
Length of the weight dual to h: 560
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: D8D^{1}_8 Containing regular semisimple subalgebra number 2: E8E^{1}_8
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V28ψ+V26ψ+2V22ψ+2V18ψ+V16ψ+3V14ψ+2V10ψ+V8ψ+V6ψ+V4ψ+V2ψV_{28\psi}+V_{26\psi}+2V_{22\psi}+2V_{18\psi}+V_{16\psi}+3V_{14\psi}+2V_{10\psi}+V_{8\psi}+V_{6\psi}+V_{4\psi}+V_{2\psi}
Below is one possible realization of the sl(2) subalgebra.
h=28h8+54h7+80h6+104h5+128h4+86h3+64h2+44h1e=14g25+28g15+26g14+54g13+36g12+28g11+50g10+44g9f=g9+g10+g11+g12+g13+g14+g15+g25\begin{array}{rcl}h&=&28h_{8}+54h_{7}+80h_{6}+104h_{5}+128h_{4}+86h_{3}+64h_{2}+44h_{1}\\ e&=&14g_{25}+28g_{15}+26g_{14}+54g_{13}+36g_{12}+28g_{11}+50g_{10}+44g_{9}\\ f&=&g_{-9}+g_{-10}+g_{-11}+g_{-12}+g_{-13}+g_{-14}+g_{-15}+g_{-25}\end{array}
Lie brackets of the above elements.
h=28h8+54h7+80h6+104h5+128h4+86h3+64h2+44h1e=14g25+28g15+26g14+54g13+36g12+28g11+50g10+44g9f=g9+g10+g11+g12+g13+g14+g15+g25\begin{array}{rcl}[e, f]&=&28h_{8}+54h_{7}+80h_{6}+104h_{5}+128h_{4}+86h_{3}+64h_{2}+44h_{1}\\ [h, e]&=&28g_{25}+56g_{15}+52g_{14}+108g_{13}+72g_{12}+56g_{11}+100g_{10}+88g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-13}-2g_{-14}-2g_{-15}-2g_{-25}\end{array}
Centralizer type: 00
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Unknown elements.
h=28h8+54h7+80h6+104h5+128h4+86h3+64h2+44h1e=x1g25+x7g15+x2g14+x6g13+x3g12+x8g11+x5g10+x4g9e=x12g9+x13g10+x16g11+x11g12+x14g13+x10g14+x15g15+x9g25\begin{array}{rcl}h&=&28h_{8}+54h_{7}+80h_{6}+104h_{5}+128h_{4}+86h_{3}+64h_{2}+44h_{1}\\ e&=&x_{1} g_{25}+x_{7} g_{15}+x_{2} g_{14}+x_{6} g_{13}+x_{3} g_{12}+x_{8} g_{11}+x_{5} g_{10}+x_{4} g_{9}\\ f&=&x_{12} g_{-9}+x_{13} g_{-10}+x_{16} g_{-11}+x_{11} g_{-12}+x_{14} g_{-13}+x_{10} g_{-14}+x_{15} g_{-15}+x_{9} g_{-25}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x7x1528)h8+(x2x10+x7x1554)h7+(x2x10+x6x1480)h6+(x1x9+x3x11+x6x14104)h5+(x1x9+x3x11+x5x13+x8x16128)h4+(x1x9+x4x12+x8x1686)h3+(x1x9+x5x1364)h2+(x4x1244)h1[e,f] - h = \left(x_{7} x_{15} -28\right)h_{8}+\left(x_{2} x_{10} +x_{7} x_{15} -54\right)h_{7}+\left(x_{2} x_{10} +x_{6} x_{14} -80\right)h_{6}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} -104\right)h_{5}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{8} x_{16} -128\right)h_{4}+\left(x_{1} x_{9} +x_{4} x_{12} +x_{8} x_{16} -86\right)h_{3}+\left(x_{1} x_{9} +x_{5} x_{13} -64\right)h_{2}+\left(x_{4} x_{12} -44\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x9+x5x1364=0x1x9+x4x12+x8x1686=0x1x9+x3x11+x5x13+x8x16128=0x1x9+x3x11+x6x14104=0x2x10+x6x1480=0x2x10+x7x1554=0x4x1244=0x7x1528=0\begin{array}{rcl}x_{1} x_{9} +x_{5} x_{13} -64&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{8} x_{16} -86&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{8} x_{16} -128&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} -104&=&0\\x_{2} x_{10} +x_{6} x_{14} -80&=&0\\x_{2} x_{10} +x_{7} x_{15} -54&=&0\\x_{4} x_{12} -44&=&0\\x_{7} x_{15} -28&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=28h8+54h7+80h6+104h5+128h4+86h3+64h2+44h1e=x1g25+x7g15+x2g14+x6g13+x3g12+x8g11+x5g10+x4g9f=g9+g10+g11+g12+g13+g14+g15+g25\begin{array}{rcl}h&=&28h_{8}+54h_{7}+80h_{6}+104h_{5}+128h_{4}+86h_{3}+64h_{2}+44h_{1}\\e&=&x_{1} g_{25}+x_{7} g_{15}+x_{2} g_{14}+x_{6} g_{13}+x_{3} g_{12}+x_{8} g_{11}+x_{5} g_{10}+x_{4} g_{9}\\f&=&g_{-9}+g_{-10}+g_{-11}+g_{-12}+g_{-13}+g_{-14}+g_{-15}+g_{-25}\end{array}
Matrix form of the system we are trying to solve: (1000100010010001101010011010010001000100010000100001000000000010)[col. vect.]=(648612810480544428)\begin{pmatrix}1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1\\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}64\\ 86\\ 128\\ 104\\ 80\\ 54\\ 44\\ 28\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=28h8+54h7+80h6+104h5+128h4+86h3+64h2+44h1e=x1g25+x7g15+x2g14+x6g13+x3g12+x8g11+x5g10+x4g9f=x12g9+x13g10+x16g11+x11g12+x14g13+x10g14+x15g15+x9g25\begin{array}{rcl}h&=&28h_{8}+54h_{7}+80h_{6}+104h_{5}+128h_{4}+86h_{3}+64h_{2}+44h_{1}\\ e&=&x_{1} g_{25}+x_{7} g_{15}+x_{2} g_{14}+x_{6} g_{13}+x_{3} g_{12}+x_{8} g_{11}+x_{5} g_{10}+x_{4} g_{9}\\ f&=&x_{12} g_{-9}+x_{13} g_{-10}+x_{16} g_{-11}+x_{11} g_{-12}+x_{14} g_{-13}+x_{10} g_{-14}+x_{15} g_{-15}+x_{9} g_{-25}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x9+x5x1364=0x1x9+x4x12+x8x1686=0x1x9+x3x11+x5x13+x8x16128=0x1x9+x3x11+x6x14104=0x2x10+x6x1480=0x2x10+x7x1554=0x4x1244=0x7x1528=0\begin{array}{rcl}x_{1} x_{9} +x_{5} x_{13} -64&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{8} x_{16} -86&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{8} x_{16} -128&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} -104&=&0\\x_{2} x_{10} +x_{6} x_{14} -80&=&0\\x_{2} x_{10} +x_{7} x_{15} -54&=&0\\x_{4} x_{12} -44&=&0\\x_{7} x_{15} -28&=&0\\\end{array}

A1232A^{232}_1
h-characteristic: (2, 0, 0, 2, 0, 0, 2, 2)
Length of the weight dual to h: 464
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: E7+A1E^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 2: E8E^{1}_8
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V26ψ+2V22ψ+V20ψ+V18ψ+2V16ψ+2V14ψ+V12ψ+3V10ψ+2V6ψ+V4ψ+2V2ψV_{26\psi}+2V_{22\psi}+V_{20\psi}+V_{18\psi}+2V_{16\psi}+2V_{14\psi}+V_{12\psi}+3V_{10\psi}+2V_{6\psi}+V_{4\psi}+2V_{2\psi}
Below is one possible realization of the sl(2) subalgebra.
h=26h8+50h7+72h6+94h5+116h4+78h3+58h2+40h1e=21g27+g26+57g25+152g21+13g17+852g14+152g12+26g8+592g4+40g1f=g1+g4+g8+g12+g14+g21+g25+g26+g27\begin{array}{rcl}h&=&26h_{8}+50h_{7}+72h_{6}+94h_{5}+116h_{4}+78h_{3}+58h_{2}+40h_{1}\\ e&=&21g_{27}+g_{26}+57g_{25}+15/2g_{21}+13g_{17}+85/2g_{14}+15/2g_{12}+26g_{8}+59/2g_{4}+40g_{1}\\ f&=&g_{-1}+g_{-4}+g_{-8}+g_{-12}+g_{-14}+g_{-21}+g_{-25}+g_{-26}+g_{-27}\end{array}
Lie brackets of the above elements.
h=26h8+50h7+72h6+94h5+116h4+78h3+58h2+40h1e=21g27+g26+57g25+152g21+13g17+852g14+152g12+26g8+592g4+40g1f=g1+g4+g8+g12+g14+g21+g25+g26+g27\begin{array}{rcl}[e, f]&=&26h_{8}+50h_{7}+72h_{6}+94h_{5}+116h_{4}+78h_{3}+58h_{2}+40h_{1}\\ [h, e]&=&42g_{27}+2g_{26}+114g_{25}+15g_{21}+26g_{17}+85g_{14}+15g_{12}+52g_{8}+59g_{4}+80g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-4}-2g_{-8}-2g_{-12}-2g_{-14}-2g_{-21}-2g_{-25}-2g_{-26}-2g_{-27}\end{array}
Centralizer type: 00
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Unknown elements.
h=26h8+50h7+72h6+94h5+116h4+78h3+58h2+40h1e=x6g27+x7g26+x4g25+x3g21+x10g17+x9g14+x2g12+x1g8+x8g4+x5g1e=x15g1+x18g4+x11g8+x12g12+x19g14+x20g17+x13g21+x14g25+x17g26+x16g27\begin{array}{rcl}h&=&26h_{8}+50h_{7}+72h_{6}+94h_{5}+116h_{4}+78h_{3}+58h_{2}+40h_{1}\\ e&=&x_{6} g_{27}+x_{7} g_{26}+x_{4} g_{25}+x_{3} g_{21}+x_{10} g_{17}+x_{9} g_{14}+x_{2} g_{12}+x_{1} g_{8}+x_{8} g_{4}+x_{5} g_{1}\\ f&=&x_{15} g_{-1}+x_{18} g_{-4}+x_{11} g_{-8}+x_{12} g_{-12}+x_{19} g_{-14}+x_{20} g_{-17}+x_{13} g_{-21}+x_{14} g_{-25}+x_{17} g_{-26}+x_{16} g_{-27}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x18+x3x19x4x20)g5+(x1x1126)h8+(x3x13+x9x1950)h7+(x3x13+x6x16+x7x17+x9x1972)h6+(x2x12+x3x13+x4x14+x6x16+x7x1794)h5+(x2x12+x4x14+x6x16+x7x17+x8x18+x10x20116)h4+(x4x14+x6x16+x10x2078)h3+(x4x14+x7x17+x10x2058)h2+(x5x1540)h1+(x8x12+x9x13x10x14)g5[e,f] - h = \left(-x_{2} x_{18} +x_{3} x_{19} -x_{4} x_{20} \right)g_{5}+\left(x_{1} x_{11} -26\right)h_{8}+\left(x_{3} x_{13} +x_{9} x_{19} -50\right)h_{7}+\left(x_{3} x_{13} +x_{6} x_{16} +x_{7} x_{17} +x_{9} x_{19} -72\right)h_{6}+\left(x_{2} x_{12} +x_{3} x_{13} +x_{4} x_{14} +x_{6} x_{16} +x_{7} x_{17} -94\right)h_{5}+\left(x_{2} x_{12} +x_{4} x_{14} +x_{6} x_{16} +x_{7} x_{17} +x_{8} x_{18} +x_{10} x_{20} -116\right)h_{4}+\left(x_{4} x_{14} +x_{6} x_{16} +x_{10} x_{20} -78\right)h_{3}+\left(x_{4} x_{14} +x_{7} x_{17} +x_{10} x_{20} -58\right)h_{2}+\left(x_{5} x_{15} -40\right)h_{1}+\left(-x_{8} x_{12} +x_{9} x_{13} -x_{10} x_{14} \right)g_{-5}
The polynomial system that corresponds to finding the h, e, f triple:
x1x1126=0x2x12+x4x14+x6x16+x7x17+x8x18+x10x20116=0x2x12+x3x13+x4x14+x6x16+x7x1794=0x2x18+x3x19x4x20=0x3x13+x6x16+x7x17+x9x1972=0x3x13+x9x1950=0x4x14+x7x17+x10x2058=0x4x14+x6x16+x10x2078=0x5x1540=0x8x12+x9x13x10x14=0\begin{array}{rcl}x_{1} x_{11} -26&=&0\\x_{2} x_{12} +x_{4} x_{14} +x_{6} x_{16} +x_{7} x_{17} +x_{8} x_{18} +x_{10} x_{20} -116&=&0\\x_{2} x_{12} +x_{3} x_{13} +x_{4} x_{14} +x_{6} x_{16} +x_{7} x_{17} -94&=&0\\-x_{2} x_{18} +x_{3} x_{19} -x_{4} x_{20} &=&0\\x_{3} x_{13} +x_{6} x_{16} +x_{7} x_{17} +x_{9} x_{19} -72&=&0\\x_{3} x_{13} +x_{9} x_{19} -50&=&0\\x_{4} x_{14} +x_{7} x_{17} +x_{10} x_{20} -58&=&0\\x_{4} x_{14} +x_{6} x_{16} +x_{10} x_{20} -78&=&0\\x_{5} x_{15} -40&=&0\\-x_{8} x_{12} +x_{9} x_{13} -x_{10} x_{14} &=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=26h8+50h7+72h6+94h5+116h4+78h3+58h2+40h1e=x6g27+x7g26+x4g25+x3g21+x10g17+x9g14+x2g12+x1g8+x8g4+x5g1f=g1+g4+g8+g12+g14+g21+g25+g26+g27\begin{array}{rcl}h&=&26h_{8}+50h_{7}+72h_{6}+94h_{5}+116h_{4}+78h_{3}+58h_{2}+40h_{1}\\e&=&x_{6} g_{27}+x_{7} g_{26}+x_{4} g_{25}+x_{3} g_{21}+x_{10} g_{17}+x_{9} g_{14}+x_{2} g_{12}+x_{1} g_{8}+x_{8} g_{4}+x_{5} g_{1}\\f&=&g_{-1}+g_{-4}+g_{-8}+g_{-12}+g_{-14}+g_{-21}+g_{-25}+g_{-26}+g_{-27}\end{array}
Matrix form of the system we are trying to solve: (1000000000010101110001110110000110000000001001101000100000100001001000000101000000001000000000000111)[col. vect.]=(2611694072505878400)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0\\ 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & -1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}26\\ 116\\ 94\\ 0\\ 72\\ 50\\ 58\\ 78\\ 40\\ 0\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=26h8+50h7+72h6+94h5+116h4+78h3+58h2+40h1e=x6g27+x7g26+x4g25+x3g21+x10g17+x9g14+x2g12+x1g8+x8g4+x5g1f=x15g1+x18g4+x11g8+x12g12+x19g14+x20g17+x13g21+x14g25+x17g26+x16g27\begin{array}{rcl}h&=&26h_{8}+50h_{7}+72h_{6}+94h_{5}+116h_{4}+78h_{3}+58h_{2}+40h_{1}\\ e&=&x_{6} g_{27}+x_{7} g_{26}+x_{4} g_{25}+x_{3} g_{21}+x_{10} g_{17}+x_{9} g_{14}+x_{2} g_{12}+x_{1} g_{8}+x_{8} g_{4}+x_{5} g_{1}\\ f&=&x_{15} g_{-1}+x_{18} g_{-4}+x_{11} g_{-8}+x_{12} g_{-12}+x_{19} g_{-14}+x_{20} g_{-17}+x_{13} g_{-21}+x_{14} g_{-25}+x_{17} g_{-26}+x_{16} g_{-27}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x1126=0x2x12+x4x14+x6x16+x7x17+x8x18+x10x20116=0x2x12+x3x13+x4x14+x6x16+x7x1794=0x2x18+x3x19x4x20=0x3x13+x6x16+x7x17+x9x1972=0x3x13+x9x1950=0x4x14+x7x17+x10x2058=0x4x14+x6x16+x10x2078=0x5x1540=0x8x12+x9x13x10x14=0\begin{array}{rcl}x_{1} x_{11} -26&=&0\\x_{2} x_{12} +x_{4} x_{14} +x_{6} x_{16} +x_{7} x_{17} +x_{8} x_{18} +x_{10} x_{20} -116&=&0\\x_{2} x_{12} +x_{3} x_{13} +x_{4} x_{14} +x_{6} x_{16} +x_{7} x_{17} -94&=&0\\-x_{2} x_{18} +x_{3} x_{19} -x_{4} x_{20} &=&0\\x_{3} x_{13} +x_{6} x_{16} +x_{7} x_{17} +x_{9} x_{19} -72&=&0\\x_{3} x_{13} +x_{9} x_{19} -50&=&0\\x_{4} x_{14} +x_{7} x_{17} +x_{10} x_{20} -58&=&0\\x_{4} x_{14} +x_{6} x_{16} +x_{10} x_{20} -78&=&0\\x_{5} x_{15} -40&=&0\\-x_{8} x_{12} +x_{9} x_{13} -x_{10} x_{14} &=&0\\\end{array}

A1231A^{231}_1
h-characteristic: (2, 1, 1, 0, 1, 0, 2, 2)
Length of the weight dual to h: 462
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: E7E^{1}_7
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V26ψ+V22ψ+2V21ψ+V18ψ+V16ψ+2V15ψ+V14ψ+2V11ψ+2V10ψ+V6ψ+2V5ψ+V2ψ+3V0V_{26\psi}+V_{22\psi}+2V_{21\psi}+V_{18\psi}+V_{16\psi}+2V_{15\psi}+V_{14\psi}+2V_{11\psi}+2V_{10\psi}+V_{6\psi}+2V_{5\psi}+V_{2\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=26h8+50h7+72h6+94h5+115h4+78h3+58h2+40h1e=572g275g26+6g19+8g187g17+5g14+26g810g7+403g1f=3g14g7+g8+2g143g17+4g18g262g27\begin{array}{rcl}h&=&26h_{8}+50h_{7}+72h_{6}+94h_{5}+115h_{4}+78h_{3}+58h_{2}+40h_{1}\\ e&=&-57/2g_{27}-5g_{26}+6g_{19}+8g_{18}-7g_{17}+5g_{14}+26g_{8}-10g_{7}+40/3g_{1}\\ f&=&3g_{-1}-4g_{-7}+g_{-8}+2g_{-14}-3g_{-17}+4g_{-18}-g_{-26}-2g_{-27}\end{array}
Lie brackets of the above elements.
h=26h8+50h7+72h6+94h5+115h4+78h3+58h2+40h1e=572g275g26+6g19+8g187g17+5g14+26g810g7+403g1f=3g14g7+g8+2g143g17+4g18g262g27\begin{array}{rcl}[e, f]&=&26h_{8}+50h_{7}+72h_{6}+94h_{5}+115h_{4}+78h_{3}+58h_{2}+40h_{1}\\ [h, e]&=&-57g_{27}-10g_{26}+12g_{19}+16g_{18}-14g_{17}+10g_{14}+52g_{8}-20g_{7}+80/3g_{1}\\ [h, f]&=&-6g_{-1}+8g_{-7}-2g_{-8}-4g_{-14}+6g_{-17}-8g_{-18}+2g_{-26}+4g_{-27}\end{array}
Centralizer type: A1A_1
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Unknown elements.
h=26h8+50h7+72h6+94h5+115h4+78h3+58h2+40h1e=x4g27+x2g26+x9g19+x7g18+x6g17+x3g14+x1g8+x8g7+x5g1e=x14g1+x17g7+x10g8+x12g14+x15g17+x16g18+x18g19+x11g26+x13g27\begin{array}{rcl}h&=&26h_{8}+50h_{7}+72h_{6}+94h_{5}+115h_{4}+78h_{3}+58h_{2}+40h_{1}\\ e&=&x_{4} g_{27}+x_{2} g_{26}+x_{9} g_{19}+x_{7} g_{18}+x_{6} g_{17}+x_{3} g_{14}+x_{1} g_{8}+x_{8} g_{7}+x_{5} g_{1}\\ f&=&x_{14} g_{-1}+x_{17} g_{-7}+x_{10} g_{-8}+x_{12} g_{-14}+x_{15} g_{-17}+x_{16} g_{-18}+x_{18} g_{-19}+x_{11} g_{-26}+x_{13} g_{-27}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x16+x3x17x4x18)g6+(x1x1026)h8+(x3x12+x8x1750)h7+(x2x11+x3x12+x4x1372)h6+(x2x11+x4x13+x7x16+x9x1894)h5+(x2x11+x4x13+x6x15+x7x16+x9x18115)h4+(x4x13+x6x15+x9x1878)h3+(x2x11+x6x15+x7x1658)h2+(x5x1440)h1+(x7x11+x8x12x9x13)g6[e,f] - h = \left(-x_{2} x_{16} +x_{3} x_{17} -x_{4} x_{18} \right)g_{6}+\left(x_{1} x_{10} -26\right)h_{8}+\left(x_{3} x_{12} +x_{8} x_{17} -50\right)h_{7}+\left(x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} -72\right)h_{6}+\left(x_{2} x_{11} +x_{4} x_{13} +x_{7} x_{16} +x_{9} x_{18} -94\right)h_{5}+\left(x_{2} x_{11} +x_{4} x_{13} +x_{6} x_{15} +x_{7} x_{16} +x_{9} x_{18} -115\right)h_{4}+\left(x_{4} x_{13} +x_{6} x_{15} +x_{9} x_{18} -78\right)h_{3}+\left(x_{2} x_{11} +x_{6} x_{15} +x_{7} x_{16} -58\right)h_{2}+\left(x_{5} x_{14} -40\right)h_{1}+\left(-x_{7} x_{11} +x_{8} x_{12} -x_{9} x_{13} \right)g_{-6}
The polynomial system that corresponds to finding the h, e, f triple:
x1x1026=0x2x11+x6x15+x7x1658=0x2x11+x4x13+x6x15+x7x16+x9x18115=0x2x11+x4x13+x7x16+x9x1894=0x2x11+x3x12+x4x1372=0x2x16+x3x17x4x18=0x3x12+x8x1750=0x4x13+x6x15+x9x1878=0x5x1440=0x7x11+x8x12x9x13=0\begin{array}{rcl}x_{1} x_{10} -26&=&0\\x_{2} x_{11} +x_{6} x_{15} +x_{7} x_{16} -58&=&0\\x_{2} x_{11} +x_{4} x_{13} +x_{6} x_{15} +x_{7} x_{16} +x_{9} x_{18} -115&=&0\\x_{2} x_{11} +x_{4} x_{13} +x_{7} x_{16} +x_{9} x_{18} -94&=&0\\x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} -72&=&0\\-x_{2} x_{16} +x_{3} x_{17} -x_{4} x_{18} &=&0\\x_{3} x_{12} +x_{8} x_{17} -50&=&0\\x_{4} x_{13} +x_{6} x_{15} +x_{9} x_{18} -78&=&0\\x_{5} x_{14} -40&=&0\\-x_{7} x_{11} +x_{8} x_{12} -x_{9} x_{13} &=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=26h8+50h7+72h6+94h5+115h4+78h3+58h2+40h1e=x4g27+x2g26+x9g19+x7g18+x6g17+x3g14+x1g8+x8g7+x5g1f=3g14g7+g8+2g143g17+4g18g262g27\begin{array}{rcl}h&=&26h_{8}+50h_{7}+72h_{6}+94h_{5}+115h_{4}+78h_{3}+58h_{2}+40h_{1}\\e&=&x_{4} g_{27}+x_{2} g_{26}+x_{9} g_{19}+x_{7} g_{18}+x_{6} g_{17}+x_{3} g_{14}+x_{1} g_{8}+x_{8} g_{7}+x_{5} g_{1}\\f&=&3g_{-1}-4g_{-7}+g_{-8}+2g_{-14}-3g_{-17}+4g_{-18}-g_{-26}-2g_{-27}\end{array}
Matrix form of the system we are trying to solve: (100000000010003400010203400010200400012200000044000000002000040000203000000030000000000122)[col. vect.]=(2658115947205078400)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & -3 & 4 & 0 & 0\\ 0 & -1 & 0 & -2 & 0 & -3 & 4 & 0 & 0\\ 0 & -1 & 0 & -2 & 0 & 0 & 4 & 0 & 0\\ 0 & -1 & 2 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & -4 & -4 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & -4 & 0\\ 0 & 0 & 0 & -2 & 0 & -3 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}26\\ 58\\ 115\\ 94\\ 72\\ 0\\ 50\\ 78\\ 40\\ 0\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=26h8+50h7+72h6+94h5+115h4+78h3+58h2+40h1e=x4g27+x2g26+x9g19+x7g18+x6g17+x3g14+x1g8+x8g7+x5g1f=x14g1+x17g7+x10g8+x12g14+x15g17+x16g18+x18g19+x11g26+x13g27\begin{array}{rcl}h&=&26h_{8}+50h_{7}+72h_{6}+94h_{5}+115h_{4}+78h_{3}+58h_{2}+40h_{1}\\ e&=&x_{4} g_{27}+x_{2} g_{26}+x_{9} g_{19}+x_{7} g_{18}+x_{6} g_{17}+x_{3} g_{14}+x_{1} g_{8}+x_{8} g_{7}+x_{5} g_{1}\\ f&=&x_{14} g_{-1}+x_{17} g_{-7}+x_{10} g_{-8}+x_{12} g_{-14}+x_{15} g_{-17}+x_{16} g_{-18}+x_{18} g_{-19}+x_{11} g_{-26}+x_{13} g_{-27}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x1026=0x2x11+x6x15+x7x1658=0x2x11+x4x13+x6x15+x7x16+x9x18115=0x2x11+x4x13+x7x16+x9x1894=0x2x11+x3x12+x4x1372=0x2x16+x3x17x4x18=0x3x12+x8x1750=0x4x13+x6x15+x9x1878=0x5x1440=0x7x11+x8x12x9x13=0\begin{array}{rcl}x_{1} x_{10} -26&=&0\\x_{2} x_{11} +x_{6} x_{15} +x_{7} x_{16} -58&=&0\\x_{2} x_{11} +x_{4} x_{13} +x_{6} x_{15} +x_{7} x_{16} +x_{9} x_{18} -115&=&0\\x_{2} x_{11} +x_{4} x_{13} +x_{7} x_{16} +x_{9} x_{18} -94&=&0\\x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} -72&=&0\\-x_{2} x_{16} +x_{3} x_{17} -x_{4} x_{18} &=&0\\x_{3} x_{12} +x_{8} x_{17} -50&=&0\\x_{4} x_{13} +x_{6} x_{15} +x_{9} x_{18} -78&=&0\\x_{5} x_{14} -40&=&0\\-x_{7} x_{11} +x_{8} x_{12} -x_{9} x_{13} &=&0\\\end{array}

A1184A^{184}_1
h-characteristic: (2, 0, 0, 2, 0, 0, 2, 0)
Length of the weight dual to h: 368
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: E8E^{1}_8 Containing regular semisimple subalgebra number 2: D8D^{1}_8
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 2V22ψ+V20ψ+V18ψ+V16ψ+3V14ψ+2V12ψ+4V10ψ+V8ψ+V6ψ+V4ψ+3V2ψ2V_{22\psi}+V_{20\psi}+V_{18\psi}+V_{16\psi}+3V_{14\psi}+2V_{12\psi}+4V_{10\psi}+V_{8\psi}+V_{6\psi}+V_{4\psi}+3V_{2\psi}
Below is one possible realization of the sl(2) subalgebra.
h=22h8+44h7+64h6+84h5+104h4+70h3+52h2+36h1e=3g33+8g29+21g27+6g26+14g2511g22+4g2112g2023g18+2g17+3g15+11g14+5g12+21g11+16g10+9g9+7g77g427g1f=g1+g4+g7+g9+g10+g11+g12+g14+g15+g17g18+g19g20+g21g22+g25g26+g27+g29g33\begin{array}{rcl}h&=&22h_{8}+44h_{7}+64h_{6}+84h_{5}+104h_{4}+70h_{3}+52h_{2}+36h_{1}\\ e&=&-3g_{33}+8g_{29}+21g_{27}+6g_{26}+14g_{25}-11g_{22}+4g_{21}-12g_{20}-23g_{18}+2g_{17}+3g_{15}+11g_{14}+5g_{12}+21g_{11}+16g_{10}+9g_{9}+7g_{7}-7g_{4}-27g_{1}\\ f&=&-g_{-1}+g_{-4}+g_{-7}+g_{-9}+g_{-10}+g_{-11}+g_{-12}+g_{-14}+g_{-15}+g_{-17}-g_{-18}+g_{-19}-g_{-20}+g_{-21}-g_{-22}+g_{-25}-g_{-26}+g_{-27}+g_{-29}-g_{-33}\end{array}
Lie brackets of the above elements.
h=22h8+44h7+64h6+84h5+104h4+70h3+52h2+36h1e=3g33+8g29+21g27+6g26+14g2511g22+4g2112g2023g18+2g17+3g15+11g14+5g12+21g11+16g10+9g9+7g77g427g1f=g1+g4+g7+g9+g10+g11+g12+g14+g15+g17g18+g19g20+g21g22+g25g26+g27+g29g33\begin{array}{rcl}[e, f]&=&22h_{8}+44h_{7}+64h_{6}+84h_{5}+104h_{4}+70h_{3}+52h_{2}+36h_{1}\\ [h, e]&=&-6g_{33}+16g_{29}+42g_{27}+12g_{26}+28g_{25}-22g_{22}+8g_{21}-24g_{20}-46g_{18}+4g_{17}+6g_{15}+22g_{14}+10g_{12}+42g_{11}+32g_{10}+18g_{9}+14g_{7}-14g_{4}-54g_{1}\\ [h, f]&=&2g_{-1}-2g_{-4}-2g_{-7}-2g_{-9}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-14}-2g_{-15}-2g_{-17}+2g_{-18}-2g_{-19}+2g_{-20}-2g_{-21}+2g_{-22}-2g_{-25}+2g_{-26}-2g_{-27}-2g_{-29}+2g_{-33}\end{array}
Centralizer type: 00
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Unknown elements.
h=22h8+44h7+64h6+84h5+104h4+70h3+52h2+36h1e=x2g33+x3g29+x5g27+x6g26+x7g25+x8g22+x9g21+x10g20+x11g19+x12g18+x13g17+x14g15+x15g14+x16g12+x17g11+x18g10+x1g9+x19g7+x20g4+x4g1e=x24g1+x40g4+x39g7+x21g9+x38g10+x37g11+x36g12+x35g14+x34g15+x33g17+x32g18+x31g19+x30g20+x29g21+x28g22+x27g25+x26g26+x25g27+x23g29+x22g33\begin{array}{rcl}h&=&22h_{8}+44h_{7}+64h_{6}+84h_{5}+104h_{4}+70h_{3}+52h_{2}+36h_{1}\\ e&=&x_{2} g_{33}+x_{3} g_{29}+x_{5} g_{27}+x_{6} g_{26}+x_{7} g_{25}+x_{8} g_{22}+x_{9} g_{21}+x_{10} g_{20}+x_{11} g_{19}+x_{12} g_{18}+x_{13} g_{17}+x_{14} g_{15}+x_{15} g_{14}+x_{16} g_{12}+x_{17} g_{11}+x_{18} g_{10}+x_{1} g_{9}+x_{19} g_{7}+x_{20} g_{4}+x_{4} g_{1}\\ f&=&x_{24} g_{-1}+x_{40} g_{-4}+x_{39} g_{-7}+x_{21} g_{-9}+x_{38} g_{-10}+x_{37} g_{-11}+x_{36} g_{-12}+x_{35} g_{-14}+x_{34} g_{-15}+x_{33} g_{-17}+x_{32} g_{-18}+x_{31} g_{-19}+x_{30} g_{-20}+x_{29} g_{-21}+x_{28} g_{-22}+x_{27} g_{-25}+x_{26} g_{-26}+x_{25} g_{-27}+x_{23} g_{-29}+x_{22} g_{-33}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x33+x3x34x5x37x6x38+x9x39x10x40)g13+(x3x29x8x35x14x39)g8+(x2x27x5x31x6x32+x8x34x10x36+x15x39)g6+(x3x28x7x33+x9x35x11x37x12x38x16x40)g5+(x1x24+x2x26+x5x30+x7x32+x11x36+x13x38+x17x40)g3+(x2x25+x6x30+x7x31+x12x36+x13x37+x18x40)g2+(x3x23+x8x28+x14x3422)h8+(x3x23+x8x28+x9x29+x14x34+x15x35+x19x3944)h7+(x2x22+x3x23+x5x25+x6x26+x8x28+x9x29+x10x30+x15x3564)h6+(x2x22+x3x23+x5x25+x6x26+x7x27+x9x29+x10x30+x11x31+x12x32+x16x3684)h5+(x2x22+x5x25+x6x26+x7x27+x10x30+x11x31+x12x32+x13x33+x16x36+x17x37+x18x38+x20x40104)h4+(x1x21+x2x22+x5x25+x7x27+x11x31+x13x33+x17x3770)h3+(x2x22+x6x26+x7x27+x12x32+x13x33+x18x3852)h2+(x1x21+x4x2436)h1+(x5x22+x10x26+x11x27+x16x32+x17x33+x20x38)g2+(x4x21+x6x22+x10x25+x12x27+x16x31+x18x33+x20x37)g3+(x8x23x13x27+x15x29x17x31x18x32x20x36)g5+(x7x22x11x25x12x26+x14x28x16x30+x19x35)g6+(x9x23x15x28x19x34)g8+(x13x22+x14x23x17x25x18x26+x19x29x20x30)g13[e,f] - h = \left(-x_{2} x_{33} +x_{3} x_{34} -x_{5} x_{37} -x_{6} x_{38} +x_{9} x_{39} -x_{10} x_{40} \right)g_{13}+\left(-x_{3} x_{29} -x_{8} x_{35} -x_{14} x_{39} \right)g_{8}+\left(-x_{2} x_{27} -x_{5} x_{31} -x_{6} x_{32} +x_{8} x_{34} -x_{10} x_{36} +x_{15} x_{39} \right)g_{6}+\left(x_{3} x_{28} -x_{7} x_{33} +x_{9} x_{35} -x_{11} x_{37} -x_{12} x_{38} -x_{16} x_{40} \right)g_{5}+\left(-x_{1} x_{24} +x_{2} x_{26} +x_{5} x_{30} +x_{7} x_{32} +x_{11} x_{36} +x_{13} x_{38} +x_{17} x_{40} \right)g_{3}+\left(x_{2} x_{25} +x_{6} x_{30} +x_{7} x_{31} +x_{12} x_{36} +x_{13} x_{37} +x_{18} x_{40} \right)g_{2}+\left(x_{3} x_{23} +x_{8} x_{28} +x_{14} x_{34} -22\right)h_{8}+\left(x_{3} x_{23} +x_{8} x_{28} +x_{9} x_{29} +x_{14} x_{34} +x_{15} x_{35} +x_{19} x_{39} -44\right)h_{7}+\left(x_{2} x_{22} +x_{3} x_{23} +x_{5} x_{25} +x_{6} x_{26} +x_{8} x_{28} +x_{9} x_{29} +x_{10} x_{30} +x_{15} x_{35} -64\right)h_{6}+\left(x_{2} x_{22} +x_{3} x_{23} +x_{5} x_{25} +x_{6} x_{26} +x_{7} x_{27} +x_{9} x_{29} +x_{10} x_{30} +x_{11} x_{31} +x_{12} x_{32} +x_{16} x_{36} -84\right)h_{5}+\left(x_{2} x_{22} +x_{5} x_{25} +x_{6} x_{26} +x_{7} x_{27} +x_{10} x_{30} +x_{11} x_{31} +x_{12} x_{32} +x_{13} x_{33} +x_{16} x_{36} +x_{17} x_{37} +x_{18} x_{38} +x_{20} x_{40} -104\right)h_{4}+\left(x_{1} x_{21} +x_{2} x_{22} +x_{5} x_{25} +x_{7} x_{27} +x_{11} x_{31} +x_{13} x_{33} +x_{17} x_{37} -70\right)h_{3}+\left(x_{2} x_{22} +x_{6} x_{26} +x_{7} x_{27} +x_{12} x_{32} +x_{13} x_{33} +x_{18} x_{38} -52\right)h_{2}+\left(x_{1} x_{21} +x_{4} x_{24} -36\right)h_{1}+\left(x_{5} x_{22} +x_{10} x_{26} +x_{11} x_{27} +x_{16} x_{32} +x_{17} x_{33} +x_{20} x_{38} \right)g_{-2}+\left(-x_{4} x_{21} +x_{6} x_{22} +x_{10} x_{25} +x_{12} x_{27} +x_{16} x_{31} +x_{18} x_{33} +x_{20} x_{37} \right)g_{-3}+\left(x_{8} x_{23} -x_{13} x_{27} +x_{15} x_{29} -x_{17} x_{31} -x_{18} x_{32} -x_{20} x_{36} \right)g_{-5}+\left(-x_{7} x_{22} -x_{11} x_{25} -x_{12} x_{26} +x_{14} x_{28} -x_{16} x_{30} +x_{19} x_{35} \right)g_{-6}+\left(-x_{9} x_{23} -x_{15} x_{28} -x_{19} x_{34} \right)g_{-8}+\left(-x_{13} x_{22} +x_{14} x_{23} -x_{17} x_{25} -x_{18} x_{26} +x_{19} x_{29} -x_{20} x_{30} \right)g_{-13}
The polynomial system that corresponds to finding the h, e, f triple:
x1x21+x4x2436=0x1x21+x2x22+x5x25+x7x27+x11x31+x13x33+x17x3770=0x1x24+x2x26+x5x30+x7x32+x11x36+x13x38+x17x40=0x2x22+x6x26+x7x27+x12x32+x13x33+x18x3852=0x2x22+x5x25+x6x26+x7x27+x10x30+x11x31+x12x32+x13x33+x16x36+x17x37+x18x38+x20x40104=0x2x22+x3x23+x5x25+x6x26+x7x27+x9x29+x10x30+x11x31+x12x32+x16x3684=0x2x22+x3x23+x5x25+x6x26+x8x28+x9x29+x10x30+x15x3564=0x2x25+x6x30+x7x31+x12x36+x13x37+x18x40=0x2x27x5x31x6x32+x8x34x10x36+x15x39=0x2x33+x3x34x5x37x6x38+x9x39x10x40=0x3x23+x8x28+x9x29+x14x34+x15x35+x19x3944=0x3x23+x8x28+x14x3422=0x3x28x7x33+x9x35x11x37x12x38x16x40=0x3x29x8x35x14x39=0x4x21+x6x22+x10x25+x12x27+x16x31+x18x33+x20x37=0x5x22+x10x26+x11x27+x16x32+x17x33+x20x38=0x7x22x11x25x12x26+x14x28x16x30+x19x35=0x8x23x13x27+x15x29x17x31x18x32x20x36=0x9x23x15x28x19x34=0x13x22+x14x23x17x25x18x26+x19x29x20x30=0\begin{array}{rcl}x_{1} x_{21} +x_{4} x_{24} -36&=&0\\x_{1} x_{21} +x_{2} x_{22} +x_{5} x_{25} +x_{7} x_{27} +x_{11} x_{31} +x_{13} x_{33} +x_{17} x_{37} -70&=&0\\-x_{1} x_{24} +x_{2} x_{26} +x_{5} x_{30} +x_{7} x_{32} +x_{11} x_{36} +x_{13} x_{38} +x_{17} x_{40} &=&0\\x_{2} x_{22} +x_{6} x_{26} +x_{7} x_{27} +x_{12} x_{32} +x_{13} x_{33} +x_{18} x_{38} -52&=&0\\x_{2} x_{22} +x_{5} x_{25} +x_{6} x_{26} +x_{7} x_{27} +x_{10} x_{30} +x_{11} x_{31} +x_{12} x_{32} +x_{13} x_{33} +x_{16} x_{36} +x_{17} x_{37} +x_{18} x_{38} +x_{20} x_{40} -104&=&0\\x_{2} x_{22} +x_{3} x_{23} +x_{5} x_{25} +x_{6} x_{26} +x_{7} x_{27} +x_{9} x_{29} +x_{10} x_{30} +x_{11} x_{31} +x_{12} x_{32} +x_{16} x_{36} -84&=&0\\x_{2} x_{22} +x_{3} x_{23} +x_{5} x_{25} +x_{6} x_{26} +x_{8} x_{28} +x_{9} x_{29} +x_{10} x_{30} +x_{15} x_{35} -64&=&0\\x_{2} x_{25} +x_{6} x_{30} +x_{7} x_{31} +x_{12} x_{36} +x_{13} x_{37} +x_{18} x_{40} &=&0\\-x_{2} x_{27} -x_{5} x_{31} -x_{6} x_{32} +x_{8} x_{34} -x_{10} x_{36} +x_{15} x_{39} &=&0\\-x_{2} x_{33} +x_{3} x_{34} -x_{5} x_{37} -x_{6} x_{38} +x_{9} x_{39} -x_{10} x_{40} &=&0\\x_{3} x_{23} +x_{8} x_{28} +x_{9} x_{29} +x_{14} x_{34} +x_{15} x_{35} +x_{19} x_{39} -44&=&0\\x_{3} x_{23} +x_{8} x_{28} +x_{14} x_{34} -22&=&0\\x_{3} x_{28} -x_{7} x_{33} +x_{9} x_{35} -x_{11} x_{37} -x_{12} x_{38} -x_{16} x_{40} &=&0\\-x_{3} x_{29} -x_{8} x_{35} -x_{14} x_{39} &=&0\\-x_{4} x_{21} +x_{6} x_{22} +x_{10} x_{25} +x_{12} x_{27} +x_{16} x_{31} +x_{18} x_{33} +x_{20} x_{37} &=&0\\x_{5} x_{22} +x_{10} x_{26} +x_{11} x_{27} +x_{16} x_{32} +x_{17} x_{33} +x_{20} x_{38} &=&0\\-x_{7} x_{22} -x_{11} x_{25} -x_{12} x_{26} +x_{14} x_{28} -x_{16} x_{30} +x_{19} x_{35} &=&0\\x_{8} x_{23} -x_{13} x_{27} +x_{15} x_{29} -x_{17} x_{31} -x_{18} x_{32} -x_{20} x_{36} &=&0\\-x_{9} x_{23} -x_{15} x_{28} -x_{19} x_{34} &=&0\\-x_{13} x_{22} +x_{14} x_{23} -x_{17} x_{25} -x_{18} x_{26} +x_{19} x_{29} -x_{20} x_{30} &=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=22h8+44h7+64h6+84h5+104h4+70h3+52h2+36h1e=x2g33+x3g29+x5g27+x6g26+x7g25+x8g22+x9g21+x10g20+x11g19+x12g18+x13g17+x14g15+x15g14+x16g12+x17g11+x18g10+x1g9+x19g7+x20g4+x4g1f=g1+g4+g7+g9+g10+g11+g12+g14+g15+g17g18+g19g20+g21g22+g25g26+g27+g29g33\begin{array}{rcl}h&=&22h_{8}+44h_{7}+64h_{6}+84h_{5}+104h_{4}+70h_{3}+52h_{2}+36h_{1}\\e&=&x_{2} g_{33}+x_{3} g_{29}+x_{5} g_{27}+x_{6} g_{26}+x_{7} g_{25}+x_{8} g_{22}+x_{9} g_{21}+x_{10} g_{20}+x_{11} g_{19}+x_{12} g_{18}+x_{13} g_{17}+x_{14} g_{15}+x_{15} g_{14}+x_{16} g_{12}+x_{17} g_{11}+x_{18} g_{10}+x_{1} g_{9}+x_{19} g_{7}+x_{20} g_{4}+x_{4} g_{1}\\f&=&-g_{-1}+g_{-4}+g_{-7}+g_{-9}+g_{-10}+g_{-11}+g_{-12}+g_{-14}+g_{-15}+g_{-17}-g_{-18}+g_{-19}-g_{-20}+g_{-21}-g_{-22}+g_{-25}-g_{-26}+g_{-27}+g_{-29}-g_{-33}\end{array}
Matrix form of the system we are trying to solve: (1001000000000000000011001010001010001000110010100010100010000100011000011000010001001110011110011101011011101111000100000110110111000010000001000110000110000100010011010100001000000110110011000000000000100001100001100010001000010000010000000010001010110001000000100001000001000000000101000101000101010000100001100001100100000010001101010010000000010000101011010000000010000010001000000000000011001111)[col. vect.]=(36700521048464000442200000000)\begin{pmatrix}1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & -1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 1 & -1 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & -1 & 0 & 0 & 1 & -1 & 1 & 0 & 0 & -1 & 1 & -1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1\\ 0 & -1 & 1 & 0 & 1 & -1 & 1 & 0 & 1 & -1 & 1 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 1 & -1 & 0 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & -1 & 0 & 0 & -1 & 1 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & -1 & -1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & -1 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 1 & 0 & -1 & 0 & 1 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & -1 & 1 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & -1 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}36\\ 70\\ 0\\ 52\\ 104\\ 84\\ 64\\ 0\\ 0\\ 0\\ 44\\ 22\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=22h8+44h7+64h6+84h5+104h4+70h3+52h2+36h1e=x2g33+x3g29+x5g27+x6g26+x7g25+x8g22+x9g21+x10g20+x11g19+x12g18+x13g17+x14g15+x15g14+x16g12+x17g11+x18g10+x1g9+x19g7+x20g4+x4g1f=x24g1+x40g4+x39g7+x21g9+x38g10+x37g11+x36g12+x35g14+x34g15+x33g17+x32g18+x31g19+x30g20+x29g21+x28g22+x27g25+x26g26+x25g27+x23g29+x22g33\begin{array}{rcl}h&=&22h_{8}+44h_{7}+64h_{6}+84h_{5}+104h_{4}+70h_{3}+52h_{2}+36h_{1}\\ e&=&x_{2} g_{33}+x_{3} g_{29}+x_{5} g_{27}+x_{6} g_{26}+x_{7} g_{25}+x_{8} g_{22}+x_{9} g_{21}+x_{10} g_{20}+x_{11} g_{19}+x_{12} g_{18}+x_{13} g_{17}+x_{14} g_{15}+x_{15} g_{14}+x_{16} g_{12}+x_{17} g_{11}+x_{18} g_{10}+x_{1} g_{9}+x_{19} g_{7}+x_{20} g_{4}+x_{4} g_{1}\\ f&=&x_{24} g_{-1}+x_{40} g_{-4}+x_{39} g_{-7}+x_{21} g_{-9}+x_{38} g_{-10}+x_{37} g_{-11}+x_{36} g_{-12}+x_{35} g_{-14}+x_{34} g_{-15}+x_{33} g_{-17}+x_{32} g_{-18}+x_{31} g_{-19}+x_{30} g_{-20}+x_{29} g_{-21}+x_{28} g_{-22}+x_{27} g_{-25}+x_{26} g_{-26}+x_{25} g_{-27}+x_{23} g_{-29}+x_{22} g_{-33}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x21+x4x2436=0x1x21+x2x22+x5x25+x7x27+x11x31+x13x33+x17x3770=0x1x24+x2x26+x5x30+x7x32+x11x36+x13x38+x17x40=0x2x22+x6x26+x7x27+x12x32+x13x33+x18x3852=0x2x22+x5x25+x6x26+x7x27+x10x30+x11x31+x12x32+x13x33+x16x36+x17x37+x18x38+x20x40104=0x2x22+x3x23+x5x25+x6x26+x7x27+x9x29+x10x30+x11x31+x12x32+x16x3684=0x2x22+x3x23+x5x25+x6x26+x8x28+x9x29+x10x30+x15x3564=0x2x25+x6x30+x7x31+x12x36+x13x37+x18x40=0x2x27x5x31x6x32+x8x34x10x36+x15x39=0x2x33+x3x34x5x37x6x38+x9x39x10x40=0x3x23+x8x28+x9x29+x14x34+x15x35+x19x3944=0x3x23+x8x28+x14x3422=0x3x28x7x33+x9x35x11x37x12x38x16x40=0x3x29x8x35x14x39=0x4x21+x6x22+x10x25+x12x27+x16x31+x18x33+x20x37=0x5x22+x10x26+x11x27+x16x32+x17x33+x20x38=0x7x22x11x25x12x26+x14x28x16x30+x19x35=0x8x23x13x27+x15x29x17x31x18x32x20x36=0x9x23x15x28x19x34=0x13x22+x14x23x17x25x18x26+x19x29x20x30=0\begin{array}{rcl}x_{1} x_{21} +x_{4} x_{24} -36&=&0\\x_{1} x_{21} +x_{2} x_{22} +x_{5} x_{25} +x_{7} x_{27} +x_{11} x_{31} +x_{13} x_{33} +x_{17} x_{37} -70&=&0\\-x_{1} x_{24} +x_{2} x_{26} +x_{5} x_{30} +x_{7} x_{32} +x_{11} x_{36} +x_{13} x_{38} +x_{17} x_{40} &=&0\\x_{2} x_{22} +x_{6} x_{26} +x_{7} x_{27} +x_{12} x_{32} +x_{13} x_{33} +x_{18} x_{38} -52&=&0\\x_{2} x_{22} +x_{5} x_{25} +x_{6} x_{26} +x_{7} x_{27} +x_{10} x_{30} +x_{11} x_{31} +x_{12} x_{32} +x_{13} x_{33} +x_{16} x_{36} +x_{17} x_{37} +x_{18} x_{38} +x_{20} x_{40} -104&=&0\\x_{2} x_{22} +x_{3} x_{23} +x_{5} x_{25} +x_{6} x_{26} +x_{7} x_{27} +x_{9} x_{29} +x_{10} x_{30} +x_{11} x_{31} +x_{12} x_{32} +x_{16} x_{36} -84&=&0\\x_{2} x_{22} +x_{3} x_{23} +x_{5} x_{25} +x_{6} x_{26} +x_{8} x_{28} +x_{9} x_{29} +x_{10} x_{30} +x_{15} x_{35} -64&=&0\\x_{2} x_{25} +x_{6} x_{30} +x_{7} x_{31} +x_{12} x_{36} +x_{13} x_{37} +x_{18} x_{40} &=&0\\-x_{2} x_{27} -x_{5} x_{31} -x_{6} x_{32} +x_{8} x_{34} -x_{10} x_{36} +x_{15} x_{39} &=&0\\-x_{2} x_{33} +x_{3} x_{34} -x_{5} x_{37} -x_{6} x_{38} +x_{9} x_{39} -x_{10} x_{40} &=&0\\x_{3} x_{23} +x_{8} x_{28} +x_{9} x_{29} +x_{14} x_{34} +x_{15} x_{35} +x_{19} x_{39} -44&=&0\\x_{3} x_{23} +x_{8} x_{28} +x_{14} x_{34} -22&=&0\\x_{3} x_{28} -x_{7} x_{33} +x_{9} x_{35} -x_{11} x_{37} -x_{12} x_{38} -x_{16} x_{40} &=&0\\-x_{3} x_{29} -x_{8} x_{35} -x_{14} x_{39} &=&0\\-x_{4} x_{21} +x_{6} x_{22} +x_{10} x_{25} +x_{12} x_{27} +x_{16} x_{31} +x_{18} x_{33} +x_{20} x_{37} &=&0\\x_{5} x_{22} +x_{10} x_{26} +x_{11} x_{27} +x_{16} x_{32} +x_{17} x_{33} +x_{20} x_{38} &=&0\\-x_{7} x_{22} -x_{11} x_{25} -x_{12} x_{26} +x_{14} x_{28} -x_{16} x_{30} +x_{19} x_{35} &=&0\\x_{8} x_{23} -x_{13} x_{27} +x_{15} x_{29} -x_{17} x_{31} -x_{18} x_{32} -x_{20} x_{36} &=&0\\-x_{9} x_{23} -x_{15} x_{28} -x_{19} x_{34} &=&0\\-x_{13} x_{22} +x_{14} x_{23} -x_{17} x_{25} -x_{18} x_{26} +x_{19} x_{29} -x_{20} x_{30} &=&0\\\end{array}

A1182A^{182}_1
h-characteristic: (2, 1, 1, 0, 1, 1, 0, 1)
Length of the weight dual to h: 364
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: D7D^{1}_7
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V22ψ+2V21ψ+V18ψ+2V15ψ+V14ψ+3V12ψ+2V11ψ+V10ψ+2V9ψ+V6ψ+2V3ψ+V2ψ+3V0V_{22\psi}+2V_{21\psi}+V_{18\psi}+2V_{15\psi}+V_{14\psi}+3V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+V_{6\psi}+2V_{3\psi}+V_{2\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=22h8+43h7+64h6+84h5+103h4+70h3+52h2+36h1e=21g28+22g22+30g19+12g18+40g17+21g13+36g1f=g1+g13+g17+g18+g19+g22+g28\begin{array}{rcl}h&=&22h_{8}+43h_{7}+64h_{6}+84h_{5}+103h_{4}+70h_{3}+52h_{2}+36h_{1}\\ e&=&21g_{28}+22g_{22}+30g_{19}+12g_{18}+40g_{17}+21g_{13}+36g_{1}\\ f&=&g_{-1}+g_{-13}+g_{-17}+g_{-18}+g_{-19}+g_{-22}+g_{-28}\end{array}
Lie brackets of the above elements.
h=22h8+43h7+64h6+84h5+103h4+70h3+52h2+36h1e=21g28+22g22+30g19+12g18+40g17+21g13+36g1f=g1+g13+g17+g18+g19+g22+g28\begin{array}{rcl}[e, f]&=&22h_{8}+43h_{7}+64h_{6}+84h_{5}+103h_{4}+70h_{3}+52h_{2}+36h_{1}\\ [h, e]&=&42g_{28}+44g_{22}+60g_{19}+24g_{18}+80g_{17}+42g_{13}+72g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-13}-2g_{-17}-2g_{-18}-2g_{-19}-2g_{-22}-2g_{-28}\end{array}
Centralizer type: A12A^{2}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 3): h7h4h_{7}-h_{4}, g4+g7g_{4}+g_{-7}, g7+g4g_{7}+g_{-4}
Basis of centralizer intersected with cartan (dimension: 1): h7h4h_{7}-h_{4}
Cartan of centralizer (dimension: 1): h7h4h_{7}-h_{4}
Cartan-generating semisimple element: h7h4h_{7}-h_{4}
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Characteristic polynomial ad H: x34xx^3-4x
Factorization of characteristic polynomial of ad H: (x )(x -2)(x +2)
Eigenvalues of ad H: 00, 22, 2-2
3 eigenvectors of ad H: 1, 0, 0(1,0,0), 0, 0, 1(0,0,1), 0, 1, 0(0,1,0)
Centralizer type: A^{2}_1
Reductive components (1 total):
Scalar product computed: (160)\begin{pmatrix}1/60\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h7h4h_{7}-h_{4}
matching e: g7+g4g_{7}+g_{-4}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h7h4h_{7}-h_{4}
matching e: g7+g4g_{7}+g_{-4}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240)\begin{pmatrix}240\\ \end{pmatrix}
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Unknown elements.
h=22h8+43h7+64h6+84h5+103h4+70h3+52h2+36h1e=x6g28+x2g22+x3g19+x1g18+x5g17+x7g13+x4g1e=x11g1+x14g13+x12g17+x8g18+x10g19+x9g22+x13g28\begin{array}{rcl}h&=&22h_{8}+43h_{7}+64h_{6}+84h_{5}+103h_{4}+70h_{3}+52h_{2}+36h_{1}\\ e&=&x_{6} g_{28}+x_{2} g_{22}+x_{3} g_{19}+x_{1} g_{18}+x_{5} g_{17}+x_{7} g_{13}+x_{4} g_{1}\\ f&=&x_{11} g_{-1}+x_{14} g_{-13}+x_{12} g_{-17}+x_{8} g_{-18}+x_{10} g_{-19}+x_{9} g_{-22}+x_{13} g_{-28}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x922)h8+(x2x9+x6x1343)h7+(x2x9+x6x13+x7x1464)h6+(x1x8+x3x10+x6x13+x7x1484)h5+(x1x8+x3x10+x5x12+x6x13103)h4+(x3x10+x5x1270)h3+(x1x8+x5x1252)h2+(x4x1136)h1[e,f] - h = \left(x_{2} x_{9} -22\right)h_{8}+\left(x_{2} x_{9} +x_{6} x_{13} -43\right)h_{7}+\left(x_{2} x_{9} +x_{6} x_{13} +x_{7} x_{14} -64\right)h_{6}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -84\right)h_{5}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -103\right)h_{4}+\left(x_{3} x_{10} +x_{5} x_{12} -70\right)h_{3}+\left(x_{1} x_{8} +x_{5} x_{12} -52\right)h_{2}+\left(x_{4} x_{11} -36\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x5x1252=0x1x8+x3x10+x5x12+x6x13103=0x1x8+x3x10+x6x13+x7x1484=0x2x9+x6x13+x7x1464=0x2x9+x6x1343=0x2x922=0x3x10+x5x1270=0x4x1136=0\begin{array}{rcl}x_{1} x_{8} +x_{5} x_{12} -52&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -103&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -84&=&0\\x_{2} x_{9} +x_{6} x_{13} +x_{7} x_{14} -64&=&0\\x_{2} x_{9} +x_{6} x_{13} -43&=&0\\x_{2} x_{9} -22&=&0\\x_{3} x_{10} +x_{5} x_{12} -70&=&0\\x_{4} x_{11} -36&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=22h8+43h7+64h6+84h5+103h4+70h3+52h2+36h1e=x6g28+x2g22+x3g19+x1g18+x5g17+x7g13+x4g1f=g1+g13+g17+g18+g19+g22+g28\begin{array}{rcl}h&=&22h_{8}+43h_{7}+64h_{6}+84h_{5}+103h_{4}+70h_{3}+52h_{2}+36h_{1}\\e&=&x_{6} g_{28}+x_{2} g_{22}+x_{3} g_{19}+x_{1} g_{18}+x_{5} g_{17}+x_{7} g_{13}+x_{4} g_{1}\\f&=&g_{-1}+g_{-13}+g_{-17}+g_{-18}+g_{-19}+g_{-22}+g_{-28}\end{array}
Matrix form of the system we are trying to solve: (10001001010110101001101000110100010010000000101000001000)[col. vect.]=(52103846443227036)\begin{pmatrix}1 & 0 & 0 & 0 & 1 & 0 & 0\\ 1 & 0 & 1 & 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}52\\ 103\\ 84\\ 64\\ 43\\ 22\\ 70\\ 36\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=22h8+43h7+64h6+84h5+103h4+70h3+52h2+36h1e=x6g28+x2g22+x3g19+x1g18+x5g17+x7g13+x4g1f=x11g1+x14g13+x12g17+x8g18+x10g19+x9g22+x13g28\begin{array}{rcl}h&=&22h_{8}+43h_{7}+64h_{6}+84h_{5}+103h_{4}+70h_{3}+52h_{2}+36h_{1}\\ e&=&x_{6} g_{28}+x_{2} g_{22}+x_{3} g_{19}+x_{1} g_{18}+x_{5} g_{17}+x_{7} g_{13}+x_{4} g_{1}\\ f&=&x_{11} g_{-1}+x_{14} g_{-13}+x_{12} g_{-17}+x_{8} g_{-18}+x_{10} g_{-19}+x_{9} g_{-22}+x_{13} g_{-28}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x5x1252=0x1x8+x3x10+x5x12+x6x13103=0x1x8+x3x10+x6x13+x7x1484=0x2x9+x6x13+x7x1464=0x2x9+x6x1343=0x2x922=0x3x10+x5x1270=0x4x1136=0\begin{array}{rcl}x_{1} x_{8} +x_{5} x_{12} -52&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -103&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -84&=&0\\x_{2} x_{9} +x_{6} x_{13} +x_{7} x_{14} -64&=&0\\x_{2} x_{9} +x_{6} x_{13} -43&=&0\\x_{2} x_{9} -22&=&0\\x_{3} x_{10} +x_{5} x_{12} -70&=&0\\x_{4} x_{11} -36&=&0\\\end{array}

A1160A^{160}_1
h-characteristic: (0, 0, 0, 2, 0, 0, 2, 2)
Length of the weight dual to h: 320
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: E6+A2E^{1}_6+A^{1}_2 Containing regular semisimple subalgebra number 2: E8E^{1}_8 Containing regular semisimple subalgebra number 3: E7+A1E^{1}_7+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V22ψ+2V18ψ+3V16ψ+3V14ψ+3V10ψ+3V8ψ+2V6ψ+V4ψ+4V2ψV_{22\psi}+2V_{18\psi}+3V_{16\psi}+3V_{14\psi}+3V_{10\psi}+3V_{8\psi}+2V_{6\psi}+V_{4\psi}+4V_{2\psi}
Below is one possible realization of the sl(2) subalgebra.
h=22h8+42h7+60h6+78h5+96h4+64h3+48h2+32h1e=16g38+2g20+30g19+30g18+2g17+16g16+42g14+22g8f=g8+g14+g16+g17+g18+g19+g20+g38\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+78h_{5}+96h_{4}+64h_{3}+48h_{2}+32h_{1}\\ e&=&16g_{38}+2g_{20}+30g_{19}+30g_{18}+2g_{17}+16g_{16}+42g_{14}+22g_{8}\\ f&=&g_{-8}+g_{-14}+g_{-16}+g_{-17}+g_{-18}+g_{-19}+g_{-20}+g_{-38}\end{array}
Lie brackets of the above elements.
h=22h8+42h7+60h6+78h5+96h4+64h3+48h2+32h1e=16g38+2g20+30g19+30g18+2g17+16g16+42g14+22g8f=g8+g14+g16+g17+g18+g19+g20+g38\begin{array}{rcl}[e, f]&=&22h_{8}+42h_{7}+60h_{6}+78h_{5}+96h_{4}+64h_{3}+48h_{2}+32h_{1}\\ [h, e]&=&32g_{38}+4g_{20}+60g_{19}+60g_{18}+4g_{17}+32g_{16}+84g_{14}+44g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-14}-2g_{-16}-2g_{-17}-2g_{-18}-2g_{-19}-2g_{-20}-2g_{-38}\end{array}
Centralizer type: 00
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Unknown elements.
h=22h8+42h7+60h6+78h5+96h4+64h3+48h2+32h1e=x1g38+x8g20+x3g19+x5g18+x7g17+x6g16+x4g14+x2g8e=x10g8+x12g14+x14g16+x15g17+x13g18+x11g19+x16g20+x9g38\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+78h_{5}+96h_{4}+64h_{3}+48h_{2}+32h_{1}\\ e&=&x_{1} g_{38}+x_{8} g_{20}+x_{3} g_{19}+x_{5} g_{18}+x_{7} g_{17}+x_{6} g_{16}+x_{4} g_{14}+x_{2} g_{8}\\ f&=&x_{10} g_{-8}+x_{12} g_{-14}+x_{14} g_{-16}+x_{15} g_{-17}+x_{13} g_{-18}+x_{11} g_{-19}+x_{16} g_{-20}+x_{9} g_{-38}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x1022)h8+(x4x1242)h7+(x1x9+x4x12+x8x1660)h6+(x1x9+x3x11+x5x13+x8x1678)h5+(x1x9+x3x11+x5x13+x6x14+x7x15+x8x1696)h4+(x1x9+x3x11+x6x14+x7x1564)h3+(x1x9+x5x13+x7x1548)h2+(x1x9+x6x1432)h1[e,f] - h = \left(x_{2} x_{10} -22\right)h_{8}+\left(x_{4} x_{12} -42\right)h_{7}+\left(x_{1} x_{9} +x_{4} x_{12} +x_{8} x_{16} -60\right)h_{6}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{8} x_{16} -78\right)h_{5}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -96\right)h_{4}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{7} x_{15} -64\right)h_{3}+\left(x_{1} x_{9} +x_{5} x_{13} +x_{7} x_{15} -48\right)h_{2}+\left(x_{1} x_{9} +x_{6} x_{14} -32\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x9+x6x1432=0x1x9+x5x13+x7x1548=0x1x9+x3x11+x6x14+x7x1564=0x1x9+x3x11+x5x13+x6x14+x7x15+x8x1696=0x1x9+x3x11+x5x13+x8x1678=0x1x9+x4x12+x8x1660=0x2x1022=0x4x1242=0\begin{array}{rcl}x_{1} x_{9} +x_{6} x_{14} -32&=&0\\x_{1} x_{9} +x_{5} x_{13} +x_{7} x_{15} -48&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{7} x_{15} -64&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -96&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{8} x_{16} -78&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{8} x_{16} -60&=&0\\x_{2} x_{10} -22&=&0\\x_{4} x_{12} -42&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=22h8+42h7+60h6+78h5+96h4+64h3+48h2+32h1e=x1g38+x8g20+x3g19+x5g18+x7g17+x6g16+x4g14+x2g8f=g8+g14+g16+g17+g18+g19+g20+g38\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+78h_{5}+96h_{4}+64h_{3}+48h_{2}+32h_{1}\\e&=&x_{1} g_{38}+x_{8} g_{20}+x_{3} g_{19}+x_{5} g_{18}+x_{7} g_{17}+x_{6} g_{16}+x_{4} g_{14}+x_{2} g_{8}\\f&=&g_{-8}+g_{-14}+g_{-16}+g_{-17}+g_{-18}+g_{-19}+g_{-20}+g_{-38}\end{array}
Matrix form of the system we are trying to solve: (1000010010001010101001101010111110101001100100010100000000010000)[col. vect.]=(3248649678602242)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0\\ 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1\\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1\\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}32\\ 48\\ 64\\ 96\\ 78\\ 60\\ 22\\ 42\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=22h8+42h7+60h6+78h5+96h4+64h3+48h2+32h1e=x1g38+x8g20+x3g19+x5g18+x7g17+x6g16+x4g14+x2g8f=x10g8+x12g14+x14g16+x15g17+x13g18+x11g19+x16g20+x9g38\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+78h_{5}+96h_{4}+64h_{3}+48h_{2}+32h_{1}\\ e&=&x_{1} g_{38}+x_{8} g_{20}+x_{3} g_{19}+x_{5} g_{18}+x_{7} g_{17}+x_{6} g_{16}+x_{4} g_{14}+x_{2} g_{8}\\ f&=&x_{10} g_{-8}+x_{12} g_{-14}+x_{14} g_{-16}+x_{15} g_{-17}+x_{13} g_{-18}+x_{11} g_{-19}+x_{16} g_{-20}+x_{9} g_{-38}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x9+x6x1432=0x1x9+x5x13+x7x1548=0x1x9+x3x11+x6x14+x7x1564=0x1x9+x3x11+x5x13+x6x14+x7x15+x8x1696=0x1x9+x3x11+x5x13+x8x1678=0x1x9+x4x12+x8x1660=0x2x1022=0x4x1242=0\begin{array}{rcl}x_{1} x_{9} +x_{6} x_{14} -32&=&0\\x_{1} x_{9} +x_{5} x_{13} +x_{7} x_{15} -48&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{7} x_{15} -64&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -96&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{8} x_{16} -78&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{8} x_{16} -60&=&0\\x_{2} x_{10} -22&=&0\\x_{4} x_{12} -42&=&0\\\end{array}

A1159A^{159}_1
h-characteristic: (0, 1, 1, 0, 1, 0, 2, 2)
Length of the weight dual to h: 318
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: E7E^{1}_7
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V22ψ+V18ψ+2V17ψ+V16ψ+2V15ψ+2V14ψ+2V10ψ+2V9ψ+V8ψ+2V7ψ+V6ψ+2V3ψ+2V2ψ+3V0V_{22\psi}+V_{18\psi}+2V_{17\psi}+V_{16\psi}+2V_{15\psi}+2V_{14\psi}+2V_{10\psi}+2V_{9\psi}+V_{8\psi}+2V_{7\psi}+V_{6\psi}+2V_{3\psi}+2V_{2\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=22h8+42h7+60h6+78h5+95h4+64h3+48h2+32h1e=16g32+16g2714g26+2g24+14g23+13g1917g18+3g17+14g14+22g8+28g7f=g7+g8+g14+g17g18+g19+g23+g24g26+g27g32\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+78h_{5}+95h_{4}+64h_{3}+48h_{2}+32h_{1}\\ e&=&-16g_{32}+16g_{27}-14g_{26}+2g_{24}+14g_{23}+13g_{19}-17g_{18}+3g_{17}+14g_{14}+22g_{8}+28g_{7}\\ f&=&g_{-7}+g_{-8}+g_{-14}+g_{-17}-g_{-18}+g_{-19}+g_{-23}+g_{-24}-g_{-26}+g_{-27}-g_{-32}\end{array}
Lie brackets of the above elements.
h=22h8+42h7+60h6+78h5+95h4+64h3+48h2+32h1e=16g32+16g2714g26+2g24+14g23+13g1917g18+3g17+14g14+22g8+28g7f=g7+g8+g14+g17g18+g19+g23+g24g26+g27g32\begin{array}{rcl}[e, f]&=&22h_{8}+42h_{7}+60h_{6}+78h_{5}+95h_{4}+64h_{3}+48h_{2}+32h_{1}\\ [h, e]&=&-32g_{32}+32g_{27}-28g_{26}+4g_{24}+28g_{23}+26g_{19}-34g_{18}+6g_{17}+28g_{14}+44g_{8}+56g_{7}\\ [h, f]&=&-2g_{-7}-2g_{-8}-2g_{-14}-2g_{-17}+2g_{-18}-2g_{-19}-2g_{-23}-2g_{-24}+2g_{-26}-2g_{-27}+2g_{-32}\end{array}
Centralizer type: A1A_1
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Unknown elements.
h=22h8+42h7+60h6+78h5+95h4+64h3+48h2+32h1e=x4g32+x9g27+x2g26+x8g24+x5g23+x11g19+x6g18+x10g17+x3g14+x1g8+x7g7e=x18g7+x12g8+x14g14+x21g17+x17g18+x22g19+x16g23+x19g24+x13g26+x20g27+x15g32\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+78h_{5}+95h_{4}+64h_{3}+48h_{2}+32h_{1}\\ e&=&x_{4} g_{32}+x_{9} g_{27}+x_{2} g_{26}+x_{8} g_{24}+x_{5} g_{23}+x_{11} g_{19}+x_{6} g_{18}+x_{10} g_{17}+x_{3} g_{14}+x_{1} g_{8}+x_{7} g_{7}\\ f&=&x_{18} g_{-7}+x_{12} g_{-8}+x_{14} g_{-14}+x_{21} g_{-17}+x_{17} g_{-18}+x_{22} g_{-19}+x_{16} g_{-23}+x_{19} g_{-24}+x_{13} g_{-26}+x_{20} g_{-27}+x_{15} g_{-32}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x17+x3x18x4x19x9x22)g6+(x4x20+x5x21+x8x22)g1+(x1x1222)h8+(x3x14+x7x1842)h7+(x2x13+x3x14+x4x15+x9x2060)h6+(x2x13+x4x15+x6x17+x8x19+x9x20+x11x2278)h5+(x2x13+x4x15+x5x16+x6x17+x8x19+x9x20+x10x21+x11x2295)h4+(x4x15+x5x16+x8x19+x9x20+x10x21+x11x2264)h3+(x2x13+x5x16+x6x17+x10x2148)h2+(x4x15+x5x16+x8x1932)h1+(x9x15+x10x16+x11x19)g1+(x6x13+x7x14x8x15x11x20)g6[e,f] - h = \left(-x_{2} x_{17} +x_{3} x_{18} -x_{4} x_{19} -x_{9} x_{22} \right)g_{6}+\left(x_{4} x_{20} +x_{5} x_{21} +x_{8} x_{22} \right)g_{1}+\left(x_{1} x_{12} -22\right)h_{8}+\left(x_{3} x_{14} +x_{7} x_{18} -42\right)h_{7}+\left(x_{2} x_{13} +x_{3} x_{14} +x_{4} x_{15} +x_{9} x_{20} -60\right)h_{6}+\left(x_{2} x_{13} +x_{4} x_{15} +x_{6} x_{17} +x_{8} x_{19} +x_{9} x_{20} +x_{11} x_{22} -78\right)h_{5}+\left(x_{2} x_{13} +x_{4} x_{15} +x_{5} x_{16} +x_{6} x_{17} +x_{8} x_{19} +x_{9} x_{20} +x_{10} x_{21} +x_{11} x_{22} -95\right)h_{4}+\left(x_{4} x_{15} +x_{5} x_{16} +x_{8} x_{19} +x_{9} x_{20} +x_{10} x_{21} +x_{11} x_{22} -64\right)h_{3}+\left(x_{2} x_{13} +x_{5} x_{16} +x_{6} x_{17} +x_{10} x_{21} -48\right)h_{2}+\left(x_{4} x_{15} +x_{5} x_{16} +x_{8} x_{19} -32\right)h_{1}+\left(x_{9} x_{15} +x_{10} x_{16} +x_{11} x_{19} \right)g_{-1}+\left(-x_{6} x_{13} +x_{7} x_{14} -x_{8} x_{15} -x_{11} x_{20} \right)g_{-6}
The polynomial system that corresponds to finding the h, e, f triple:
x1x1222=0x2x13+x5x16+x6x17+x10x2148=0x2x13+x4x15+x5x16+x6x17+x8x19+x9x20+x10x21+x11x2295=0x2x13+x4x15+x6x17+x8x19+x9x20+x11x2278=0x2x13+x3x14+x4x15+x9x2060=0x2x17+x3x18x4x19x9x22=0x3x14+x7x1842=0x4x15+x5x16+x8x1932=0x4x15+x5x16+x8x19+x9x20+x10x21+x11x2264=0x4x20+x5x21+x8x22=0x6x13+x7x14x8x15x11x20=0x9x15+x10x16+x11x19=0\begin{array}{rcl}x_{1} x_{12} -22&=&0\\x_{2} x_{13} +x_{5} x_{16} +x_{6} x_{17} +x_{10} x_{21} -48&=&0\\x_{2} x_{13} +x_{4} x_{15} +x_{5} x_{16} +x_{6} x_{17} +x_{8} x_{19} +x_{9} x_{20} +x_{10} x_{21} +x_{11} x_{22} -95&=&0\\x_{2} x_{13} +x_{4} x_{15} +x_{6} x_{17} +x_{8} x_{19} +x_{9} x_{20} +x_{11} x_{22} -78&=&0\\x_{2} x_{13} +x_{3} x_{14} +x_{4} x_{15} +x_{9} x_{20} -60&=&0\\-x_{2} x_{17} +x_{3} x_{18} -x_{4} x_{19} -x_{9} x_{22} &=&0\\x_{3} x_{14} +x_{7} x_{18} -42&=&0\\x_{4} x_{15} +x_{5} x_{16} +x_{8} x_{19} -32&=&0\\x_{4} x_{15} +x_{5} x_{16} +x_{8} x_{19} +x_{9} x_{20} +x_{10} x_{21} +x_{11} x_{22} -64&=&0\\x_{4} x_{20} +x_{5} x_{21} +x_{8} x_{22} &=&0\\-x_{6} x_{13} +x_{7} x_{14} -x_{8} x_{15} -x_{11} x_{20} &=&0\\x_{9} x_{15} +x_{10} x_{16} +x_{11} x_{19} &=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=22h8+42h7+60h6+78h5+95h4+64h3+48h2+32h1e=x4g32+x9g27+x2g26+x8g24+x5g23+x11g19+x6g18+x10g17+x3g14+x1g8+x7g7f=g7+g8+g14+g17g18+g19+g23+g24g26+g27g32\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+78h_{5}+95h_{4}+64h_{3}+48h_{2}+32h_{1}\\e&=&x_{4} g_{32}+x_{9} g_{27}+x_{2} g_{26}+x_{8} g_{24}+x_{5} g_{23}+x_{11} g_{19}+x_{6} g_{18}+x_{10} g_{17}+x_{3} g_{14}+x_{1} g_{8}+x_{7} g_{7}\\f&=&g_{-7}+g_{-8}+g_{-14}+g_{-17}-g_{-18}+g_{-19}+g_{-23}+g_{-24}-g_{-26}+g_{-27}-g_{-32}\end{array}
Matrix form of the system we are trying to solve: (100000000000100110001001011101111010101011010111000010001110000100001000100000001100100000011001111000110010000000011100100000000111)[col. vect.]=(22489578600423264000)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 1 & 0\\ 0 & -1 & 0 & -1 & 1 & -1 & 0 & 1 & 1 & 1 & 1\\ 0 & -1 & 0 & -1 & 0 & -1 & 0 & 1 & 1 & 0 & 1\\ 0 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 1 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 1 & 0 & 0 & 1 & 1 & 1 & 1\\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}22\\ 48\\ 95\\ 78\\ 60\\ 0\\ 42\\ 32\\ 64\\ 0\\ 0\\ 0\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=22h8+42h7+60h6+78h5+95h4+64h3+48h2+32h1e=x4g32+x9g27+x2g26+x8g24+x5g23+x11g19+x6g18+x10g17+x3g14+x1g8+x7g7f=x18g7+x12g8+x14g14+x21g17+x17g18+x22g19+x16g23+x19g24+x13g26+x20g27+x15g32\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+78h_{5}+95h_{4}+64h_{3}+48h_{2}+32h_{1}\\ e&=&x_{4} g_{32}+x_{9} g_{27}+x_{2} g_{26}+x_{8} g_{24}+x_{5} g_{23}+x_{11} g_{19}+x_{6} g_{18}+x_{10} g_{17}+x_{3} g_{14}+x_{1} g_{8}+x_{7} g_{7}\\ f&=&x_{18} g_{-7}+x_{12} g_{-8}+x_{14} g_{-14}+x_{21} g_{-17}+x_{17} g_{-18}+x_{22} g_{-19}+x_{16} g_{-23}+x_{19} g_{-24}+x_{13} g_{-26}+x_{20} g_{-27}+x_{15} g_{-32}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x1222=0x2x13+x5x16+x6x17+x10x2148=0x2x13+x4x15+x5x16+x6x17+x8x19+x9x20+x10x21+x11x2295=0x2x13+x4x15+x6x17+x8x19+x9x20+x11x2278=0x2x13+x3x14+x4x15+x9x2060=0x2x17+x3x18x4x19x9x22=0x3x14+x7x1842=0x4x15+x5x16+x8x1932=0x4x15+x5x16+x8x19+x9x20+x10x21+x11x2264=0x4x20+x5x21+x8x22=0x6x13+x7x14x8x15x11x20=0x9x15+x10x16+x11x19=0\begin{array}{rcl}x_{1} x_{12} -22&=&0\\x_{2} x_{13} +x_{5} x_{16} +x_{6} x_{17} +x_{10} x_{21} -48&=&0\\x_{2} x_{13} +x_{4} x_{15} +x_{5} x_{16} +x_{6} x_{17} +x_{8} x_{19} +x_{9} x_{20} +x_{10} x_{21} +x_{11} x_{22} -95&=&0\\x_{2} x_{13} +x_{4} x_{15} +x_{6} x_{17} +x_{8} x_{19} +x_{9} x_{20} +x_{11} x_{22} -78&=&0\\x_{2} x_{13} +x_{3} x_{14} +x_{4} x_{15} +x_{9} x_{20} -60&=&0\\-x_{2} x_{17} +x_{3} x_{18} -x_{4} x_{19} -x_{9} x_{22} &=&0\\x_{3} x_{14} +x_{7} x_{18} -42&=&0\\x_{4} x_{15} +x_{5} x_{16} +x_{8} x_{19} -32&=&0\\x_{4} x_{15} +x_{5} x_{16} +x_{8} x_{19} +x_{9} x_{20} +x_{10} x_{21} +x_{11} x_{22} -64&=&0\\x_{4} x_{20} +x_{5} x_{21} +x_{8} x_{22} &=&0\\-x_{6} x_{13} +x_{7} x_{14} -x_{8} x_{15} -x_{11} x_{20} &=&0\\x_{9} x_{15} +x_{10} x_{16} +x_{11} x_{19} &=&0\\\end{array}

A1157A^{157}_1
h-characteristic: (1, 0, 0, 1, 0, 1, 2, 2)
Length of the weight dual to h: 314
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: E6+A1E^{1}_6+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V22ψ+2V17ψ+3V16ψ+2V15ψ+V14ψ+V10ψ+2V9ψ+3V8ψ+2V7ψ+2V2ψ+4Vψ+3V0V_{22\psi}+2V_{17\psi}+3V_{16\psi}+2V_{15\psi}+V_{14\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+2V_{7\psi}+2V_{2\psi}+4V_{\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=22h8+42h7+60h6+77h5+94h4+63h3+47h2+32h1e=g31+16g30+30g27+30g26+16g16+22g8+42g7f=g7+g8+g16+g26+g27+g30+g31\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+77h_{5}+94h_{4}+63h_{3}+47h_{2}+32h_{1}\\ e&=&g_{31}+16g_{30}+30g_{27}+30g_{26}+16g_{16}+22g_{8}+42g_{7}\\ f&=&g_{-7}+g_{-8}+g_{-16}+g_{-26}+g_{-27}+g_{-30}+g_{-31}\end{array}
Lie brackets of the above elements.
h=22h8+42h7+60h6+77h5+94h4+63h3+47h2+32h1e=g31+16g30+30g27+30g26+16g16+22g8+42g7f=g7+g8+g16+g26+g27+g30+g31\begin{array}{rcl}[e, f]&=&22h_{8}+42h_{7}+60h_{6}+77h_{5}+94h_{4}+63h_{3}+47h_{2}+32h_{1}\\ [h, e]&=&2g_{31}+32g_{30}+60g_{27}+60g_{26}+32g_{16}+44g_{8}+84g_{7}\\ [h, f]&=&-2g_{-7}-2g_{-8}-2g_{-16}-2g_{-26}-2g_{-27}-2g_{-30}-2g_{-31}\end{array}
Centralizer type: A13A^{3}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 3): h5h3h2h_{5}-h_{3}-h_{2}, g3g2g5g_{3}-g_{2}-g_{-5}, g5+g2g3g_{5}+g_{-2}-g_{-3}
Basis of centralizer intersected with cartan (dimension: 1): h5h3h2h_{5}-h_{3}-h_{2}
Cartan of centralizer (dimension: 1): h5h3h2h_{5}-h_{3}-h_{2}
Cartan-generating semisimple element: h5h3h2h_{5}-h_{3}-h_{2}
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Characteristic polynomial ad H: x34xx^3-4x
Factorization of characteristic polynomial of ad H: (x )(x -2)(x +2)
Eigenvalues of ad H: 00, 22, 2-2
3 eigenvectors of ad H: 1, 0, 0(1,0,0), 0, 0, 1(0,0,1), 0, 1, 0(0,1,0)
Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed: (190)\begin{pmatrix}1/90\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h5h3h2h_{5}-h_{3}-h_{2}
matching e: g5+g2g3g_{5}+g_{-2}-g_{-3}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h5h3h2h_{5}-h_{3}-h_{2}
matching e: g5+g2g3g_{5}+g_{-2}-g_{-3}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (360)\begin{pmatrix}360\\ \end{pmatrix}
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Unknown elements.
h=22h8+42h7+60h6+77h5+94h4+63h3+47h2+32h1e=x7g31+x1g30+x3g27+x5g26+x6g16+x2g8+x4g7e=x11g7+x9g8+x13g16+x12g26+x10g27+x8g30+x14g31\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+77h_{5}+94h_{4}+63h_{3}+47h_{2}+32h_{1}\\ e&=&x_{7} g_{31}+x_{1} g_{30}+x_{3} g_{27}+x_{5} g_{26}+x_{6} g_{16}+x_{2} g_{8}+x_{4} g_{7}\\ f&=&x_{11} g_{-7}+x_{9} g_{-8}+x_{13} g_{-16}+x_{12} g_{-26}+x_{10} g_{-27}+x_{8} g_{-30}+x_{14} g_{-31}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x922)h8+(x4x1142)h7+(x3x10+x5x1260)h6+(x1x8+x3x10+x5x12+x7x1477)h5+(x1x8+x3x10+x5x12+x6x13+2x7x1494)h4+(x1x8+x3x10+x6x13+x7x1463)h3+(x1x8+x5x12+x7x1447)h2+(x1x8+x6x1332)h1[e,f] - h = \left(x_{2} x_{9} -22\right)h_{8}+\left(x_{4} x_{11} -42\right)h_{7}+\left(x_{3} x_{10} +x_{5} x_{12} -60\right)h_{6}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -77\right)h_{5}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -94\right)h_{4}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -63\right)h_{3}+\left(x_{1} x_{8} +x_{5} x_{12} +x_{7} x_{14} -47\right)h_{2}+\left(x_{1} x_{8} +x_{6} x_{13} -32\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x6x1332=0x1x8+x5x12+x7x1447=0x1x8+x3x10+x6x13+x7x1463=0x1x8+x3x10+x5x12+x6x13+2x7x1494=0x1x8+x3x10+x5x12+x7x1477=0x2x922=0x3x10+x5x1260=0x4x1142=0\begin{array}{rcl}x_{1} x_{8} +x_{6} x_{13} -32&=&0\\x_{1} x_{8} +x_{5} x_{12} +x_{7} x_{14} -47&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -63&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -94&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -77&=&0\\x_{2} x_{9} -22&=&0\\x_{3} x_{10} +x_{5} x_{12} -60&=&0\\x_{4} x_{11} -42&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=22h8+42h7+60h6+77h5+94h4+63h3+47h2+32h1e=x7g31+x1g30+x3g27+x5g26+x6g16+x2g8+x4g7f=g7+g8+g16+g26+g27+g30+g31\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+77h_{5}+94h_{4}+63h_{3}+47h_{2}+32h_{1}\\e&=&x_{7} g_{31}+x_{1} g_{30}+x_{3} g_{27}+x_{5} g_{26}+x_{6} g_{16}+x_{2} g_{8}+x_{4} g_{7}\\f&=&g_{-7}+g_{-8}+g_{-16}+g_{-26}+g_{-27}+g_{-30}+g_{-31}\end{array}
Matrix form of the system we are trying to solve: (10000101000101101001110101121010101010000000101000001000)[col. vect.]=(3247639477226042)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0 & 0 & 1 & 1\\ 1 & 0 & 1 & 0 & 1 & 1 & 2\\ 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}32\\ 47\\ 63\\ 94\\ 77\\ 22\\ 60\\ 42\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=22h8+42h7+60h6+77h5+94h4+63h3+47h2+32h1e=x7g31+x1g30+x3g27+x5g26+x6g16+x2g8+x4g7f=x11g7+x9g8+x13g16+x12g26+x10g27+x8g30+x14g31\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+77h_{5}+94h_{4}+63h_{3}+47h_{2}+32h_{1}\\ e&=&x_{7} g_{31}+x_{1} g_{30}+x_{3} g_{27}+x_{5} g_{26}+x_{6} g_{16}+x_{2} g_{8}+x_{4} g_{7}\\ f&=&x_{11} g_{-7}+x_{9} g_{-8}+x_{13} g_{-16}+x_{12} g_{-26}+x_{10} g_{-27}+x_{8} g_{-30}+x_{14} g_{-31}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x6x1332=0x1x8+x5x12+x7x1447=0x1x8+x3x10+x6x13+x7x1463=0x1x8+x3x10+x5x12+x6x13+2x7x1494=0x1x8+x3x10+x5x12+x7x1477=0x2x922=0x3x10+x5x1260=0x4x1142=0\begin{array}{rcl}x_{1} x_{8} +x_{6} x_{13} -32&=&0\\x_{1} x_{8} +x_{5} x_{12} +x_{7} x_{14} -47&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -63&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -94&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -77&=&0\\x_{2} x_{9} -22&=&0\\x_{3} x_{10} +x_{5} x_{12} -60&=&0\\x_{4} x_{11} -42&=&0\\\end{array}

A1156A^{156}_1
h-characteristic: (2, 0, 0, 0, 0, 2, 2, 2)
Length of the weight dual to h: 312
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: E6E^{1}_6
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V22ψ+7V16ψ+V14ψ+V10ψ+7V8ψ+V2ψ+14V0V_{22\psi}+7V_{16\psi}+V_{14\psi}+V_{10\psi}+7V_{8\psi}+V_{2\psi}+14V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=22h8+42h7+60h6+76h5+92h4+62h3+46h2+32h1e=16g44+30g27+30g26+22g8+42g7+16g1f=g1+g7+g8+g26+g27+g44\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+76h_{5}+92h_{4}+62h_{3}+46h_{2}+32h_{1}\\ e&=&16g_{44}+30g_{27}+30g_{26}+22g_{8}+42g_{7}+16g_{1}\\ f&=&g_{-1}+g_{-7}+g_{-8}+g_{-26}+g_{-27}+g_{-44}\end{array}
Lie brackets of the above elements.
h=22h8+42h7+60h6+76h5+92h4+62h3+46h2+32h1e=16g44+30g27+30g26+22g8+42g7+16g1f=g1+g7+g8+g26+g27+g44\begin{array}{rcl}[e, f]&=&22h_{8}+42h_{7}+60h_{6}+76h_{5}+92h_{4}+62h_{3}+46h_{2}+32h_{1}\\ [h, e]&=&32g_{44}+60g_{27}+60g_{26}+44g_{8}+84g_{7}+32g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-7}-2g_{-8}-2g_{-26}-2g_{-27}-2g_{-44}\end{array}
Centralizer type: G2G_2
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 14): g12g_{-12}, g5g_{-5}, g4g_{-4}, h4h_{4}, h5h_{5}, g3g2+g31g_{3}-g_{2}+g_{-31}, g4g_{4}, g5g_{5}, g11g10+g25g_{11}-g_{10}+g_{-25}, g12g_{12}, g17+g18g19g_{17}+g_{-18}-g_{-19}, g19g18g17g_{19}-g_{18}-g_{-17}, g25g10+g11g_{25}-g_{-10}+g_{-11}, g31g2+g3g_{31}-g_{-2}+g_{-3}
Basis of centralizer intersected with cartan (dimension: 2): h4-h_{4}, h5-h_{5}
Cartan of centralizer (dimension: 2): h4-h_{4}, h5-h_{5}
Cartan-generating semisimple element: 9h5h4-9h_{5}-h_{4}
adjoint action: (10000000000000001700000000000000700000000000000000000000000000000000000000000100000000000000700000000000000170000000000000080000000000000010000000000000009000000000000009000000000000008000000000000001)\begin{pmatrix}10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 17 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -17 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Characteristic polynomial ad H: x14584x12+117238x1010757692x8+464605361x67795026724x4+7341062400x2x^{14}-584x^{12}+117238x^{10}-10757692x^8+464605361x^6-7795026724x^4+7341062400x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -17)(x -10)(x -9)(x -8)(x -7)(x -1)(x +1)(x +7)(x +8)(x +9)(x +10)(x +17)
Eigenvalues of ad H: 00, 1717, 1010, 99, 88, 77, 11, 1-1, 7-7, 8-8, 9-9, 10-10, 17-17
14 eigenvectors of ad H: 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0,0,0,0,0), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,0,0,0,0,1), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,0,0,0,0,1,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,1,0,0,0,0,0,0)
Centralizer type: G^{1}_2
Reductive components (1 total):
Scalar product computed: (130160160190)\begin{pmatrix}1/30 & -1/60\\ -1/60 & 1/90\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
h4h_{4}
matching e: g4g_{4}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (1000000000000001000000000000002000000000000000000000000000000000000000000001000000000000002000000000000001000000000000001000000000000001000000000000000000000000000000000000000000001000000000000001)\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
h52h4-h_{5}-2h_{4}
matching e: g3g2+g31g_{3}-g_{2}+g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (3000000000000000000000000000003000000000000000000000000000000000000000000002000000000000003000000000000000000000000000001000000000000003000000000000001000000000000001000000000000001000000000000002)\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h4h_{4}
matching e: g4g_{4}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (1000000000000001000000000000002000000000000000000000000000000000000000000001000000000000002000000000000001000000000000001000000000000001000000000000000000000000000000000000000000001000000000000001)\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
h52h4-h_{5}-2h_{4}
matching e: g3g2+g31g_{3}-g_{2}+g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (3000000000000000000000000000003000000000000000000000000000000000000000000002000000000000003000000000000000000000000000001000000000000003000000000000001000000000000001000000000000001000000000000002)\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
Elements in Cartan dual to root system: (2, 1), (-2, -1), (1, 1), (-1, -1), (3, 2), (-3, -2), (3, 1), (-3, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120180180360)\begin{pmatrix}120 & -180\\ -180 & 360\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=22h8+42h7+60h6+76h5+92h4+62h3+46h2+32h1e=x1g44+x5g27+x3g26+x2g8+x4g7+x6g1e=x12g1+x10g7+x8g8+x9g26+x11g27+x7g44\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+76h_{5}+92h_{4}+62h_{3}+46h_{2}+32h_{1}\\ e&=&x_{1} g_{44}+x_{5} g_{27}+x_{3} g_{26}+x_{2} g_{8}+x_{4} g_{7}+x_{6} g_{1}\\ f&=&x_{12} g_{-1}+x_{10} g_{-7}+x_{8} g_{-8}+x_{9} g_{-26}+x_{11} g_{-27}+x_{7} g_{-44}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x822)h8+(x4x1042)h7+(x3x9+x5x1160)h6+(x1x7+x3x9+x5x1176)h5+(2x1x7+x3x9+x5x1192)h4+(2x1x7+x5x1162)h3+(x1x7+x3x946)h2+(x1x7+x6x1232)h1[e,f] - h = \left(x_{2} x_{8} -22\right)h_{8}+\left(x_{4} x_{10} -42\right)h_{7}+\left(x_{3} x_{9} +x_{5} x_{11} -60\right)h_{6}+\left(x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -76\right)h_{5}+\left(2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -92\right)h_{4}+\left(2x_{1} x_{7} +x_{5} x_{11} -62\right)h_{3}+\left(x_{1} x_{7} +x_{3} x_{9} -46\right)h_{2}+\left(x_{1} x_{7} +x_{6} x_{12} -32\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x7+x6x1232=0x1x7+x3x946=02x1x7+x5x1162=02x1x7+x3x9+x5x1192=0x1x7+x3x9+x5x1176=0x2x822=0x3x9+x5x1160=0x4x1042=0\begin{array}{rcl}x_{1} x_{7} +x_{6} x_{12} -32&=&0\\x_{1} x_{7} +x_{3} x_{9} -46&=&0\\2x_{1} x_{7} +x_{5} x_{11} -62&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -92&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -76&=&0\\x_{2} x_{8} -22&=&0\\x_{3} x_{9} +x_{5} x_{11} -60&=&0\\x_{4} x_{10} -42&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=22h8+42h7+60h6+76h5+92h4+62h3+46h2+32h1e=x1g44+x5g27+x3g26+x2g8+x4g7+x6g1f=g1+g7+g8+g26+g27+g44\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+76h_{5}+92h_{4}+62h_{3}+46h_{2}+32h_{1}\\e&=&x_{1} g_{44}+x_{5} g_{27}+x_{3} g_{26}+x_{2} g_{8}+x_{4} g_{7}+x_{6} g_{1}\\f&=&g_{-1}+g_{-7}+g_{-8}+g_{-26}+g_{-27}+g_{-44}\end{array}
Matrix form of the system we are trying to solve: (100001101000200010201010101010010000001010000100)[col. vect.]=(3246629276226042)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 1\\ 1 & 0 & 1 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 1 & 0\\ 2 & 0 & 1 & 0 & 1 & 0\\ 1 & 0 & 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}32\\ 46\\ 62\\ 92\\ 76\\ 22\\ 60\\ 42\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=22h8+42h7+60h6+76h5+92h4+62h3+46h2+32h1e=x1g44+x5g27+x3g26+x2g8+x4g7+x6g1f=x12g1+x10g7+x8g8+x9g26+x11g27+x7g44\begin{array}{rcl}h&=&22h_{8}+42h_{7}+60h_{6}+76h_{5}+92h_{4}+62h_{3}+46h_{2}+32h_{1}\\ e&=&x_{1} g_{44}+x_{5} g_{27}+x_{3} g_{26}+x_{2} g_{8}+x_{4} g_{7}+x_{6} g_{1}\\ f&=&x_{12} g_{-1}+x_{10} g_{-7}+x_{8} g_{-8}+x_{9} g_{-26}+x_{11} g_{-27}+x_{7} g_{-44}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x7+x6x1232=0x1x7+x3x946=02x1x7+x5x1162=02x1x7+x3x9+x5x1192=0x1x7+x3x9+x5x1176=0x2x822=0x3x9+x5x1160=0x4x1042=0\begin{array}{rcl}x_{1} x_{7} +x_{6} x_{12} -32&=&0\\x_{1} x_{7} +x_{3} x_{9} -46&=&0\\2x_{1} x_{7} +x_{5} x_{11} -62&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -92&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -76&=&0\\x_{2} x_{8} -22&=&0\\x_{3} x_{9} +x_{5} x_{11} -60&=&0\\x_{4} x_{10} -42&=&0\\\end{array}

A1120A^{120}_1
h-characteristic: (0, 0, 0, 2, 0, 0, 2, 0)
Length of the weight dual to h: 240
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: A8A^{1}_8 Containing regular semisimple subalgebra number 2: E8E^{1}_8 Containing regular semisimple subalgebra number 3: D8D^{1}_8
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 2V18ψ+V16ψ+3V14ψ+3V12ψ+3V10ψ+3V8ψ+5V6ψ+V4ψ+3V2ψ2V_{18\psi}+V_{16\psi}+3V_{14\psi}+3V_{12\psi}+3V_{10\psi}+3V_{8\psi}+5V_{6\psi}+V_{4\psi}+3V_{2\psi}
Below is one possible realization of the sl(2) subalgebra.
h=18h8+36h7+52h6+68h5+84h4+56h3+42h2+28h1e=8g38+18g22+18g21+8g20+14g19+20g18+14g17+20g16f=g16+g17+g18+g19+g20+g21+g22+g38\begin{array}{rcl}h&=&18h_{8}+36h_{7}+52h_{6}+68h_{5}+84h_{4}+56h_{3}+42h_{2}+28h_{1}\\ e&=&8g_{38}+18g_{22}+18g_{21}+8g_{20}+14g_{19}+20g_{18}+14g_{17}+20g_{16}\\ f&=&g_{-16}+g_{-17}+g_{-18}+g_{-19}+g_{-20}+g_{-21}+g_{-22}+g_{-38}\end{array}
Lie brackets of the above elements.
h=18h8+36h7+52h6+68h5+84h4+56h3+42h2+28h1e=8g38+18g22+18g21+8g20+14g19+20g18+14g17+20g16f=g16+g17+g18+g19+g20+g21+g22+g38\begin{array}{rcl}[e, f]&=&18h_{8}+36h_{7}+52h_{6}+68h_{5}+84h_{4}+56h_{3}+42h_{2}+28h_{1}\\ [h, e]&=&16g_{38}+36g_{22}+36g_{21}+16g_{20}+28g_{19}+40g_{18}+28g_{17}+40g_{16}\\ [h, f]&=&-2g_{-16}-2g_{-17}-2g_{-18}-2g_{-19}-2g_{-20}-2g_{-21}-2g_{-22}-2g_{-38}\end{array}
Centralizer type: 00
Unfold the hidden panel for more information.

Unknown elements.
h=18h8+36h7+52h6+68h5+84h4+56h3+42h2+28h1e=x1g38+x3g22+x6g21+x8g20+x2g19+x4g18+x7g17+x5g16e=x13g16+x15g17+x12g18+x10g19+x16g20+x14g21+x11g22+x9g38\begin{array}{rcl}h&=&18h_{8}+36h_{7}+52h_{6}+68h_{5}+84h_{4}+56h_{3}+42h_{2}+28h_{1}\\ e&=&x_{1} g_{38}+x_{3} g_{22}+x_{6} g_{21}+x_{8} g_{20}+x_{2} g_{19}+x_{4} g_{18}+x_{7} g_{17}+x_{5} g_{16}\\ f&=&x_{13} g_{-16}+x_{15} g_{-17}+x_{12} g_{-18}+x_{10} g_{-19}+x_{16} g_{-20}+x_{14} g_{-21}+x_{11} g_{-22}+x_{9} g_{-38}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x3x1118)h8+(x3x11+x6x1436)h7+(x1x9+x3x11+x6x14+x8x1652)h6+(x1x9+x2x10+x4x12+x6x14+x8x1668)h5+(x1x9+x2x10+x4x12+x5x13+x7x15+x8x1684)h4+(x1x9+x2x10+x5x13+x7x1556)h3+(x1x9+x4x12+x7x1542)h2+(x1x9+x5x1328)h1[e,f] - h = \left(x_{3} x_{11} -18\right)h_{8}+\left(x_{3} x_{11} +x_{6} x_{14} -36\right)h_{7}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{8} x_{16} -52\right)h_{6}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{6} x_{14} +x_{8} x_{16} -68\right)h_{5}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -84\right)h_{4}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{7} x_{15} -56\right)h_{3}+\left(x_{1} x_{9} +x_{4} x_{12} +x_{7} x_{15} -42\right)h_{2}+\left(x_{1} x_{9} +x_{5} x_{13} -28\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x9+x5x1328=0x1x9+x4x12+x7x1542=0x1x9+x2x10+x5x13+x7x1556=0x1x9+x2x10+x4x12+x5x13+x7x15+x8x1684=0x1x9+x2x10+x4x12+x6x14+x8x1668=0x1x9+x3x11+x6x14+x8x1652=0x3x11+x6x1436=0x3x1118=0\begin{array}{rcl}x_{1} x_{9} +x_{5} x_{13} -28&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{7} x_{15} -42&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{7} x_{15} -56&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -84&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{6} x_{14} +x_{8} x_{16} -68&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{8} x_{16} -52&=&0\\x_{3} x_{11} +x_{6} x_{14} -36&=&0\\x_{3} x_{11} -18&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=18h8+36h7+52h6+68h5+84h4+56h3+42h2+28h1e=x1g38+x3g22+x6g21+x8g20+x2g19+x4g18+x7g17+x5g16f=g16+g17+g18+g19+g20+g21+g22+g38\begin{array}{rcl}h&=&18h_{8}+36h_{7}+52h_{6}+68h_{5}+84h_{4}+56h_{3}+42h_{2}+28h_{1}\\e&=&x_{1} g_{38}+x_{3} g_{22}+x_{6} g_{21}+x_{8} g_{20}+x_{2} g_{19}+x_{4} g_{18}+x_{7} g_{17}+x_{5} g_{16}\\f&=&g_{-16}+g_{-17}+g_{-18}+g_{-19}+g_{-20}+g_{-21}+g_{-22}+g_{-38}\end{array}
Matrix form of the system we are trying to solve: (1000100010010010110010101101101111010101101001010010010000100000)[col. vect.]=(2842568468523618)\begin{pmatrix}1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0\\ 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0\\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}28\\ 42\\ 56\\ 84\\ 68\\ 52\\ 36\\ 18\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=18h8+36h7+52h6+68h5+84h4+56h3+42h2+28h1e=x1g38+x3g22+x6g21+x8g20+x2g19+x4g18+x7g17+x5g16f=x13g16+x15g17+x12g18+x10g19+x16g20+x14g21+x11g22+x9g38\begin{array}{rcl}h&=&18h_{8}+36h_{7}+52h_{6}+68h_{5}+84h_{4}+56h_{3}+42h_{2}+28h_{1}\\ e&=&x_{1} g_{38}+x_{3} g_{22}+x_{6} g_{21}+x_{8} g_{20}+x_{2} g_{19}+x_{4} g_{18}+x_{7} g_{17}+x_{5} g_{16}\\ f&=&x_{13} g_{-16}+x_{15} g_{-17}+x_{12} g_{-18}+x_{10} g_{-19}+x_{16} g_{-20}+x_{14} g_{-21}+x_{11} g_{-22}+x_{9} g_{-38}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x9+x5x1328=0x1x9+x4x12+x7x1542=0x1x9+x2x10+x5x13+x7x1556=0x1x9+x2x10+x4x12+x5x13+x7x15+x8x1684=0x1x9+x2x10+x4x12+x6x14+x8x1668=0x1x9+x3x11+x6x14+x8x1652=0x3x11+x6x1436=0x3x1118=0\begin{array}{rcl}x_{1} x_{9} +x_{5} x_{13} -28&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{7} x_{15} -42&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{7} x_{15} -56&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -84&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{6} x_{14} +x_{8} x_{16} -68&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{8} x_{16} -52&=&0\\x_{3} x_{11} +x_{6} x_{14} -36&=&0\\x_{3} x_{11} -18&=&0\\\end{array}

A1112A^{112}_1
h-characteristic: (2, 0, 0, 0, 2, 0, 0, 2)
Length of the weight dual to h: 224
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: D6+2A1D^{1}_6+2A^{1}_1 Containing regular semisimple subalgebra number 2: E7+A1E^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 3: D7D^{1}_7
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V18ψ+2V16ψ+3V14ψ+V12ψ+6V10ψ+3V8ψ+3V6ψ+2V4ψ+4V2ψ+V0V_{18\psi}+2V_{16\psi}+3V_{14\psi}+V_{12\psi}+6V_{10\psi}+3V_{8\psi}+3V_{6\psi}+2V_{4\psi}+4V_{2\psi}+V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=18h8+34h7+50h6+66h5+80h4+54h3+40h2+28h1e=10g48+15g27+g26+g25+28g23+24g21+15g12+18g8f=g8+g12+g21+g23+g25+g26+g27+g48\begin{array}{rcl}h&=&18h_{8}+34h_{7}+50h_{6}+66h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\\ e&=&10g_{48}+15g_{27}+g_{26}+g_{25}+28g_{23}+24g_{21}+15g_{12}+18g_{8}\\ f&=&g_{-8}+g_{-12}+g_{-21}+g_{-23}+g_{-25}+g_{-26}+g_{-27}+g_{-48}\end{array}
Lie brackets of the above elements.
h=18h8+34h7+50h6+66h5+80h4+54h3+40h2+28h1e=10g48+15g27+g26+g25+28g23+24g21+15g12+18g8f=g8+g12+g21+g23+g25+g26+g27+g48\begin{array}{rcl}[e, f]&=&18h_{8}+34h_{7}+50h_{6}+66h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\\ [h, e]&=&20g_{48}+30g_{27}+2g_{26}+2g_{25}+56g_{23}+48g_{21}+30g_{12}+36g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-12}-2g_{-21}-2g_{-23}-2g_{-25}-2g_{-26}-2g_{-27}-2g_{-48}\end{array}
Centralizer type: 00
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Unknown elements.
h=18h8+34h7+50h6+66h5+80h4+54h3+40h2+28h1e=x1g48+x5g27+x7g26+x8g25+x4g23+x3g21+x6g12+x2g8e=x10g8+x14g12+x11g21+x12g23+x16g25+x15g26+x13g27+x9g48\begin{array}{rcl}h&=&18h_{8}+34h_{7}+50h_{6}+66h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\\ e&=&x_{1} g_{48}+x_{5} g_{27}+x_{7} g_{26}+x_{8} g_{25}+x_{4} g_{23}+x_{3} g_{21}+x_{6} g_{12}+x_{2} g_{8}\\ f&=&x_{10} g_{-8}+x_{14} g_{-12}+x_{11} g_{-21}+x_{12} g_{-23}+x_{16} g_{-25}+x_{15} g_{-26}+x_{13} g_{-27}+x_{9} g_{-48}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x1018)h8+(x1x9+x3x1134)h7+(x1x9+x3x11+x5x13+x7x1550)h6+(x1x9+x3x11+x5x13+x6x14+x7x15+x8x1666)h5+(2x1x9+x4x12+x5x13+x6x14+x7x15+x8x1680)h4+(x1x9+x4x12+x5x13+x8x1654)h3+(x1x9+x4x12+x7x15+x8x1640)h2+(x4x1228)h1[e,f] - h = \left(x_{2} x_{10} -18\right)h_{8}+\left(x_{1} x_{9} +x_{3} x_{11} -34\right)h_{7}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} -50\right)h_{6}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -66\right)h_{5}+\left(2x_{1} x_{9} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -80\right)h_{4}+\left(x_{1} x_{9} +x_{4} x_{12} +x_{5} x_{13} +x_{8} x_{16} -54\right)h_{3}+\left(x_{1} x_{9} +x_{4} x_{12} +x_{7} x_{15} +x_{8} x_{16} -40\right)h_{2}+\left(x_{4} x_{12} -28\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x9+x4x12+x7x15+x8x1640=0x1x9+x4x12+x5x13+x8x1654=02x1x9+x4x12+x5x13+x6x14+x7x15+x8x1680=0x1x9+x3x11+x5x13+x6x14+x7x15+x8x1666=0x1x9+x3x11+x5x13+x7x1550=0x1x9+x3x1134=0x2x1018=0x4x1228=0\begin{array}{rcl}x_{1} x_{9} +x_{4} x_{12} +x_{7} x_{15} +x_{8} x_{16} -40&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{5} x_{13} +x_{8} x_{16} -54&=&0\\2x_{1} x_{9} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -80&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -66&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} -50&=&0\\x_{1} x_{9} +x_{3} x_{11} -34&=&0\\x_{2} x_{10} -18&=&0\\x_{4} x_{12} -28&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=18h8+34h7+50h6+66h5+80h4+54h3+40h2+28h1e=x1g48+x5g27+x7g26+x8g25+x4g23+x3g21+x6g12+x2g8f=g8+g12+g21+g23+g25+g26+g27+g48\begin{array}{rcl}h&=&18h_{8}+34h_{7}+50h_{6}+66h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\\e&=&x_{1} g_{48}+x_{5} g_{27}+x_{7} g_{26}+x_{8} g_{25}+x_{4} g_{23}+x_{3} g_{21}+x_{6} g_{12}+x_{2} g_{8}\\f&=&g_{-8}+g_{-12}+g_{-21}+g_{-23}+g_{-25}+g_{-26}+g_{-27}+g_{-48}\end{array}
Matrix form of the system we are trying to solve: (1001001110011001200111111010111110101010101000000100000000010000)[col. vect.]=(4054806650341828)\begin{pmatrix}1 & 0 & 0 & 1 & 0 & 0 & 1 & 1\\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1\\ 2 & 0 & 0 & 1 & 1 & 1 & 1 & 1\\ 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1\\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0\\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}40\\ 54\\ 80\\ 66\\ 50\\ 34\\ 18\\ 28\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=18h8+34h7+50h6+66h5+80h4+54h3+40h2+28h1e=x1g48+x5g27+x7g26+x8g25+x4g23+x3g21+x6g12+x2g8f=x10g8+x14g12+x11g21+x12g23+x16g25+x15g26+x13g27+x9g48\begin{array}{rcl}h&=&18h_{8}+34h_{7}+50h_{6}+66h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\\ e&=&x_{1} g_{48}+x_{5} g_{27}+x_{7} g_{26}+x_{8} g_{25}+x_{4} g_{23}+x_{3} g_{21}+x_{6} g_{12}+x_{2} g_{8}\\ f&=&x_{10} g_{-8}+x_{14} g_{-12}+x_{11} g_{-21}+x_{12} g_{-23}+x_{16} g_{-25}+x_{15} g_{-26}+x_{13} g_{-27}+x_{9} g_{-48}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x9+x4x12+x7x15+x8x1640=0x1x9+x4x12+x5x13+x8x1654=02x1x9+x4x12+x5x13+x6x14+x7x15+x8x1680=0x1x9+x3x11+x5x13+x6x14+x7x15+x8x1666=0x1x9+x3x11+x5x13+x7x1550=0x1x9+x3x1134=0x2x1018=0x4x1228=0\begin{array}{rcl}x_{1} x_{9} +x_{4} x_{12} +x_{7} x_{15} +x_{8} x_{16} -40&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{5} x_{13} +x_{8} x_{16} -54&=&0\\2x_{1} x_{9} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -80&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -66&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} -50&=&0\\x_{1} x_{9} +x_{3} x_{11} -34&=&0\\x_{2} x_{10} -18&=&0\\x_{4} x_{12} -28&=&0\\\end{array}

A1111A^{111}_1
h-characteristic: (2, 0, 0, 1, 0, 1, 0, 2)
Length of the weight dual to h: 222
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: D6+A1D^{1}_6+A^{1}_1 Containing regular semisimple subalgebra number 2: E7E^{1}_7
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V18ψ+V16ψ+2V15ψ+2V14ψ+2V11ψ+3V10ψ+4V9ψ+V8ψ+2V6ψ+2V5ψ+V4ψ+2V2ψ+2Vψ+3V0V_{18\psi}+V_{16\psi}+2V_{15\psi}+2V_{14\psi}+2V_{11\psi}+3V_{10\psi}+4V_{9\psi}+V_{8\psi}+2V_{6\psi}+2V_{5\psi}+V_{4\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=18h8+34h7+50h6+65h5+80h4+54h3+40h2+28h1e=10g41+15g31+24g28+g27+15g26+28g9+18g8f=g8+g9+g26+g27+g28+g31+g41\begin{array}{rcl}h&=&18h_{8}+34h_{7}+50h_{6}+65h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\\ e&=&10g_{41}+15g_{31}+24g_{28}+g_{27}+15g_{26}+28g_{9}+18g_{8}\\ f&=&g_{-8}+g_{-9}+g_{-26}+g_{-27}+g_{-28}+g_{-31}+g_{-41}\end{array}
Lie brackets of the above elements.
h=18h8+34h7+50h6+65h5+80h4+54h3+40h2+28h1e=10g41+15g31+24g28+g27+15g26+28g9+18g8f=g8+g9+g26+g27+g28+g31+g41\begin{array}{rcl}[e, f]&=&18h_{8}+34h_{7}+50h_{6}+65h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\\ [h, e]&=&20g_{41}+30g_{31}+48g_{28}+2g_{27}+30g_{26}+56g_{9}+36g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-9}-2g_{-26}-2g_{-27}-2g_{-28}-2g_{-31}-2g_{-41}\end{array}
Centralizer type: A1A_1
Unfold the hidden panel for more information.

Unknown elements.
h=18h8+34h7+50h6+65h5+80h4+54h3+40h2+28h1e=x1g41+x6g31+x3g28+x7g27+x5g26+x4g9+x2g8e=x9g8+x11g9+x12g26+x14g27+x10g28+x13g31+x8g41\begin{array}{rcl}h&=&18h_{8}+34h_{7}+50h_{6}+65h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\\ e&=&x_{1} g_{41}+x_{6} g_{31}+x_{3} g_{28}+x_{7} g_{27}+x_{5} g_{26}+x_{4} g_{9}+x_{2} g_{8}\\ f&=&x_{9} g_{-8}+x_{11} g_{-9}+x_{12} g_{-26}+x_{14} g_{-27}+x_{10} g_{-28}+x_{13} g_{-31}+x_{8} g_{-41}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x918)h8+(x1x8+x3x1034)h7+(x1x8+x3x10+x5x12+x7x1450)h6+(x1x8+x3x10+x5x12+x6x13+x7x1465)h5+(x1x8+x3x10+x5x12+2x6x13+x7x1480)h4+(x1x8+x4x11+x6x13+x7x1454)h3+(x1x8+x5x12+x6x1340)h2+(x4x1128)h1[e,f] - h = \left(x_{2} x_{9} -18\right)h_{8}+\left(x_{1} x_{8} +x_{3} x_{10} -34\right)h_{7}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -50\right)h_{6}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -65\right)h_{5}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -80\right)h_{4}+\left(x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -54\right)h_{3}+\left(x_{1} x_{8} +x_{5} x_{12} +x_{6} x_{13} -40\right)h_{2}+\left(x_{4} x_{11} -28\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x5x12+x6x1340=0x1x8+x4x11+x6x13+x7x1454=0x1x8+x3x10+x5x12+2x6x13+x7x1480=0x1x8+x3x10+x5x12+x6x13+x7x1465=0x1x8+x3x10+x5x12+x7x1450=0x1x8+x3x1034=0x2x918=0x4x1128=0\begin{array}{rcl}x_{1} x_{8} +x_{5} x_{12} +x_{6} x_{13} -40&=&0\\x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -54&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -80&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -65&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -50&=&0\\x_{1} x_{8} +x_{3} x_{10} -34&=&0\\x_{2} x_{9} -18&=&0\\x_{4} x_{11} -28&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=18h8+34h7+50h6+65h5+80h4+54h3+40h2+28h1e=x1g41+x6g31+x3g28+x7g27+x5g26+x4g9+x2g8f=g8+g9+g26+g27+g28+g31+g41\begin{array}{rcl}h&=&18h_{8}+34h_{7}+50h_{6}+65h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\\e&=&x_{1} g_{41}+x_{6} g_{31}+x_{3} g_{28}+x_{7} g_{27}+x_{5} g_{26}+x_{4} g_{9}+x_{2} g_{8}\\f&=&g_{-8}+g_{-9}+g_{-26}+g_{-27}+g_{-28}+g_{-31}+g_{-41}\end{array}
Matrix form of the system we are trying to solve: (10001101001011101012110101111010101101000001000000001000)[col. vect.]=(4054806550341828)\begin{pmatrix}1 & 0 & 0 & 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 1 & 0 & 1 & 1\\ 1 & 0 & 1 & 0 & 1 & 2 & 1\\ 1 & 0 & 1 & 0 & 1 & 1 & 1\\ 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}40\\ 54\\ 80\\ 65\\ 50\\ 34\\ 18\\ 28\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=18h8+34h7+50h6+65h5+80h4+54h3+40h2+28h1e=x1g41+x6g31+x3g28+x7g27+x5g26+x4g9+x2g8f=x9g8+x11g9+x12g26+x14g27+x10g28+x13g31+x8g41\begin{array}{rcl}h&=&18h_{8}+34h_{7}+50h_{6}+65h_{5}+80h_{4}+54h_{3}+40h_{2}+28h_{1}\\ e&=&x_{1} g_{41}+x_{6} g_{31}+x_{3} g_{28}+x_{7} g_{27}+x_{5} g_{26}+x_{4} g_{9}+x_{2} g_{8}\\ f&=&x_{9} g_{-8}+x_{11} g_{-9}+x_{12} g_{-26}+x_{14} g_{-27}+x_{10} g_{-28}+x_{13} g_{-31}+x_{8} g_{-41}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x5x12+x6x1340=0x1x8+x4x11+x6x13+x7x1454=0x1x8+x3x10+x5x12+2x6x13+x7x1480=0x1x8+x3x10+x5x12+x6x13+x7x1465=0x1x8+x3x10+x5x12+x7x1450=0x1x8+x3x1034=0x2x918=0x4x1128=0\begin{array}{rcl}x_{1} x_{8} +x_{5} x_{12} +x_{6} x_{13} -40&=&0\\x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -54&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -80&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -65&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -50&=&0\\x_{1} x_{8} +x_{3} x_{10} -34&=&0\\x_{2} x_{9} -18&=&0\\x_{4} x_{11} -28&=&0\\\end{array}

A1110A^{110}_1
h-characteristic: (2, 1, 1, 0, 0, 0, 1, 2)
Length of the weight dual to h: 220
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: D6D^{1}_6
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V18ψ+4V15ψ+V14ψ+6V10ψ+4V9ψ+V6ψ+4V5ψ+V2ψ+10V0V_{18\psi}+4V_{15\psi}+V_{14\psi}+6V_{10\psi}+4V_{9\psi}+V_{6\psi}+4V_{5\psi}+V_{2\psi}+10V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=18h8+34h7+49h6+64h5+79h4+54h3+40h2+28h1e=15g46+24g35+10g34+15g17+18g8+28g1f=g1+g8+g17+g34+g35+g46\begin{array}{rcl}h&=&18h_{8}+34h_{7}+49h_{6}+64h_{5}+79h_{4}+54h_{3}+40h_{2}+28h_{1}\\ e&=&15g_{46}+24g_{35}+10g_{34}+15g_{17}+18g_{8}+28g_{1}\\ f&=&g_{-1}+g_{-8}+g_{-17}+g_{-34}+g_{-35}+g_{-46}\end{array}
Lie brackets of the above elements.
h=18h8+34h7+49h6+64h5+79h4+54h3+40h2+28h1e=15g46+24g35+10g34+15g17+18g8+28g1f=g1+g8+g17+g34+g35+g46\begin{array}{rcl}[e, f]&=&18h_{8}+34h_{7}+49h_{6}+64h_{5}+79h_{4}+54h_{3}+40h_{2}+28h_{1}\\ [h, e]&=&30g_{46}+48g_{35}+20g_{34}+30g_{17}+36g_{8}+56g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-8}-2g_{-17}-2g_{-34}-2g_{-35}-2g_{-46}\end{array}
Centralizer type: B2B_2
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 10): g6g_{-6}, g4g_{-4}, h4h_{4}, h6h_{6}, g4g_{4}, g5+g20g_{5}+g_{-20}, g6g_{6}, g12g13g_{12}-g_{-13}, g13g12g_{13}-g_{-12}, g20+g5g_{20}+g_{-5}
Basis of centralizer intersected with cartan (dimension: 2): h6-h_{6}, h4-h_{4}
Cartan of centralizer (dimension: 2): h4-h_{4}, h6-h_{6}
Cartan-generating semisimple element: h69h4-h_{6}-9h_{4}
adjoint action: (20000000000180000000000000000000000000000000018000000000010000000000020000000000800000000008000000000010)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10\\ \end{pmatrix}
Characteristic polynomial ad H: x10492x8+61488x62311744x4+8294400x2x^{10}-492x^8+61488x^6-2311744x^4+8294400x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -18)(x -10)(x -8)(x -2)(x +2)(x +8)(x +10)(x +18)
Eigenvalues of ad H: 00, 1818, 1010, 88, 22, 2-2, 8-8, 10-10, 18-18
10 eigenvectors of ad H: 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,1,0), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,1,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,1), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0)
Centralizer type: B^{1}_2
Reductive components (1 total):
Scalar product computed: (160160160130)\begin{pmatrix}1/60 & -1/60\\ -1/60 & 1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
h6h4h_{6}-h_{4}
matching e: g13g12g_{13}-g_{-12}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2000000000020000000000000000000000000000000020000000000000000000002000000000020000000000200000000000)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h6-h_{6}
matching e: g6g_{-6}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2000000000000000000000000000000000000000000000000000000100000000002000000000010000000000100000000001)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h6h4h_{6}-h_{4}
matching e: g13g12g_{13}-g_{-12}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2000000000020000000000000000000000000000000020000000000000000000002000000000020000000000200000000000)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h6-h_{6}
matching e: g6g_{-6}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2000000000000000000000000000000000000000000000000000000100000000002000000000010000000000100000000001)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240120120120)\begin{pmatrix}240 & -120\\ -120 & 120\\ \end{pmatrix}
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Unknown elements.
h=18h8+34h7+49h6+64h5+79h4+54h3+40h2+28h1e=x5g46+x3g35+x1g34+x6g17+x2g8+x4g1e=x10g1+x8g8+x12g17+x7g34+x9g35+x11g46\begin{array}{rcl}h&=&18h_{8}+34h_{7}+49h_{6}+64h_{5}+79h_{4}+54h_{3}+40h_{2}+28h_{1}\\ e&=&x_{5} g_{46}+x_{3} g_{35}+x_{1} g_{34}+x_{6} g_{17}+x_{2} g_{8}+x_{4} g_{1}\\ f&=&x_{10} g_{-1}+x_{8} g_{-8}+x_{12} g_{-17}+x_{7} g_{-34}+x_{9} g_{-35}+x_{11} g_{-46}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x818)h8+(x1x7+x3x934)h7+(x1x7+x3x9+x5x1149)h6+(x1x7+x3x9+2x5x1164)h5+(x1x7+x3x9+2x5x11+x6x1279)h4+(x3x9+x5x11+x6x1254)h3+(x1x7+x5x11+x6x1240)h2+(x4x1028)h1[e,f] - h = \left(x_{2} x_{8} -18\right)h_{8}+\left(x_{1} x_{7} +x_{3} x_{9} -34\right)h_{7}+\left(x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -49\right)h_{6}+\left(x_{1} x_{7} +x_{3} x_{9} +2x_{5} x_{11} -64\right)h_{5}+\left(x_{1} x_{7} +x_{3} x_{9} +2x_{5} x_{11} +x_{6} x_{12} -79\right)h_{4}+\left(x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -54\right)h_{3}+\left(x_{1} x_{7} +x_{5} x_{11} +x_{6} x_{12} -40\right)h_{2}+\left(x_{4} x_{10} -28\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x7+x5x11+x6x1240=0x1x7+x3x9+2x5x11+x6x1279=0x1x7+x3x9+2x5x1164=0x1x7+x3x9+x5x1149=0x1x7+x3x934=0x2x818=0x3x9+x5x11+x6x1254=0x4x1028=0\begin{array}{rcl}x_{1} x_{7} +x_{5} x_{11} +x_{6} x_{12} -40&=&0\\x_{1} x_{7} +x_{3} x_{9} +2x_{5} x_{11} +x_{6} x_{12} -79&=&0\\x_{1} x_{7} +x_{3} x_{9} +2x_{5} x_{11} -64&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -49&=&0\\x_{1} x_{7} +x_{3} x_{9} -34&=&0\\x_{2} x_{8} -18&=&0\\x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -54&=&0\\x_{4} x_{10} -28&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=18h8+34h7+49h6+64h5+79h4+54h3+40h2+28h1e=x5g46+x3g35+x1g34+x6g17+x2g8+x4g1f=g1+g8+g17+g34+g35+g46\begin{array}{rcl}h&=&18h_{8}+34h_{7}+49h_{6}+64h_{5}+79h_{4}+54h_{3}+40h_{2}+28h_{1}\\e&=&x_{5} g_{46}+x_{3} g_{35}+x_{1} g_{34}+x_{6} g_{17}+x_{2} g_{8}+x_{4} g_{1}\\f&=&g_{-1}+g_{-8}+g_{-17}+g_{-34}+g_{-35}+g_{-46}\end{array}
Matrix form of the system we are trying to solve: (100011101021101020101010101000010000001011000100)[col. vect.]=(4079644934185428)\begin{pmatrix}1 & 0 & 0 & 0 & 1 & 1\\ 1 & 0 & 1 & 0 & 2 & 1\\ 1 & 0 & 1 & 0 & 2 & 0\\ 1 & 0 & 1 & 0 & 1 & 0\\ 1 & 0 & 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 1 & 1\\ 0 & 0 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}40\\ 79\\ 64\\ 49\\ 34\\ 18\\ 54\\ 28\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=18h8+34h7+49h6+64h5+79h4+54h3+40h2+28h1e=x5g46+x3g35+x1g34+x6g17+x2g8+x4g1f=x10g1+x8g8+x12g17+x7g34+x9g35+x11g46\begin{array}{rcl}h&=&18h_{8}+34h_{7}+49h_{6}+64h_{5}+79h_{4}+54h_{3}+40h_{2}+28h_{1}\\ e&=&x_{5} g_{46}+x_{3} g_{35}+x_{1} g_{34}+x_{6} g_{17}+x_{2} g_{8}+x_{4} g_{1}\\ f&=&x_{10} g_{-1}+x_{8} g_{-8}+x_{12} g_{-17}+x_{7} g_{-34}+x_{9} g_{-35}+x_{11} g_{-46}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x7+x5x11+x6x1240=0x1x7+x3x9+2x5x11+x6x1279=0x1x7+x3x9+2x5x1164=0x1x7+x3x9+x5x1149=0x1x7+x3x934=0x2x818=0x3x9+x5x11+x6x1254=0x4x1028=0\begin{array}{rcl}x_{1} x_{7} +x_{5} x_{11} +x_{6} x_{12} -40&=&0\\x_{1} x_{7} +x_{3} x_{9} +2x_{5} x_{11} +x_{6} x_{12} -79&=&0\\x_{1} x_{7} +x_{3} x_{9} +2x_{5} x_{11} -64&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -49&=&0\\x_{1} x_{7} +x_{3} x_{9} -34&=&0\\x_{2} x_{8} -18&=&0\\x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -54&=&0\\x_{4} x_{10} -28&=&0\\\end{array}

A188A^{88}_1
h-characteristic: (0, 0, 0, 2, 0, 0, 0, 2)
Length of the weight dual to h: 176
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: E8E^{1}_8 Containing regular semisimple subalgebra number 2: D8D^{1}_8 Containing regular semisimple subalgebra number 3: E6+A2E^{1}_6+A^{1}_2
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V16ψ+3V14ψ+2V12ψ+6V10ψ+3V8ψ+5V6ψ+4V4ψ+4V2ψV_{16\psi}+3V_{14\psi}+2V_{12\psi}+6V_{10\psi}+3V_{8\psi}+5V_{6\psi}+4V_{4\psi}+4V_{2\psi}
Below is one possible realization of the sl(2) subalgebra.
h=16h8+30h7+44h6+58h5+72h4+48h3+36h2+24h1e=3g47+7g417g40+g382g35+2g34+7g33+7g32+10g30+8g29+4g28+6g276g268g254g24+5g23+8g228g202g1912g1815g17+3g1612g15+16g12+12g11+2g10+4g8f=g4+g10+g11+g12+g16+g22+g23+g24+g28+g29+g30+g32+g33+g34+g35+g41+g47\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+58h_{5}+72h_{4}+48h_{3}+36h_{2}+24h_{1}\\ e&=&3g_{47}+7g_{41}-7g_{40}+g_{38}-2g_{35}+2g_{34}+7g_{33}+7g_{32}+10g_{30}+8g_{29}+4g_{28}+6g_{27}-6g_{26}-8g_{25}-4g_{24}+5g_{23}+8g_{22}-8g_{20}-2g_{19}-12g_{18}-15g_{17}+3g_{16}-12g_{15}+16g_{12}+12g_{11}+2g_{10}+4g_{8}\\ f&=&g_{-4}+g_{-10}+g_{-11}+g_{-12}+g_{-16}+g_{-22}+g_{-23}+g_{-24}+g_{-28}+g_{-29}+g_{-30}+g_{-32}+g_{-33}+g_{-34}+g_{-35}+g_{-41}+g_{-47}\end{array}
Lie brackets of the above elements.
h=16h8+30h7+44h6+58h5+72h4+48h3+36h2+24h1e=3g47+7g417g40+g382g35+2g34+7g33+7g32+10g30+8g29+4g28+6g276g268g254g24+5g23+8g228g202g1912g1815g17+3g1612g15+16g12+12g11+2g10+4g8f=g4+g10+g11+g12+g16+g22+g23+g24+g28+g29+g30+g32+g33+g34+g35+g41+g47\begin{array}{rcl}[e, f]&=&16h_{8}+30h_{7}+44h_{6}+58h_{5}+72h_{4}+48h_{3}+36h_{2}+24h_{1}\\ [h, e]&=&6g_{47}+14g_{41}-14g_{40}+2g_{38}-4g_{35}+4g_{34}+14g_{33}+14g_{32}+20g_{30}+16g_{29}+8g_{28}+12g_{27}-12g_{26}-16g_{25}-8g_{24}+10g_{23}+16g_{22}-16g_{20}-4g_{19}-24g_{18}-30g_{17}+6g_{16}-24g_{15}+32g_{12}+24g_{11}+4g_{10}+8g_{8}\\ [h, f]&=&-2g_{-4}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-16}-2g_{-22}-2g_{-23}-2g_{-24}-2g_{-28}-2g_{-29}-2g_{-30}-2g_{-32}-2g_{-33}-2g_{-34}-2g_{-35}-2g_{-41}-2g_{-47}\end{array}
Centralizer type: 00
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Unknown elements.
h=16h8+30h7+44h6+58h5+72h4+48h3+36h2+24h1e=x7g47+x1g41+x13g40+x14g38+x3g35+x4g34+x5g33+x20g32+x21g30+x2g29+x8g28+x9g27+x10g26+x11g25+x25g24+x26g23+x6g22+x15g20+x16g19+x17g18+x18g17+x28g16+x12g15+x22g12+x23g11+x24g10+x19g8+x27g4e=x55g4+x47g8+x52g10+x51g11+x50g12+x40g15+x56g16+x46g17+x45g18+x44g19+x43g20+x34g22+x54g23+x53g24+x39g25+x38g26+x37g27+x36g28+x30g29+x49g30+x48g32+x33g33+x32g34+x31g35+x42g38+x41g40+x29g41+x35g47\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+58h_{5}+72h_{4}+48h_{3}+36h_{2}+24h_{1}\\ e&=&x_{7} g_{47}+x_{1} g_{41}+x_{13} g_{40}+x_{14} g_{38}+x_{3} g_{35}+x_{4} g_{34}+x_{5} g_{33}+x_{20} g_{32}+x_{21} g_{30}+x_{2} g_{29}+x_{8} g_{28}+x_{9} g_{27}+x_{10} g_{26}+x_{11} g_{25}+x_{25} g_{24}+x_{26} g_{23}+x_{6} g_{22}+x_{15} g_{20}+x_{16} g_{19}+x_{17} g_{18}+x_{18} g_{17}+x_{28} g_{16}+x_{12} g_{15}+x_{22} g_{12}+x_{23} g_{11}+x_{24} g_{10}+x_{19} g_{8}+x_{27} g_{4}\\ f&=&x_{55} g_{-4}+x_{47} g_{-8}+x_{52} g_{-10}+x_{51} g_{-11}+x_{50} g_{-12}+x_{40} g_{-15}+x_{56} g_{-16}+x_{46} g_{-17}+x_{45} g_{-18}+x_{44} g_{-19}+x_{43} g_{-20}+x_{34} g_{-22}+x_{54} g_{-23}+x_{53} g_{-24}+x_{39} g_{-25}+x_{38} g_{-26}+x_{37} g_{-27}+x_{36} g_{-28}+x_{30} g_{-29}+x_{49} g_{-30}+x_{48} g_{-32}+x_{33} g_{-33}+x_{32} g_{-34}+x_{31} g_{-35}+x_{42} g_{-38}+x_{41} g_{-40}+x_{29} g_{-41}+x_{35} g_{-47}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x46+x2x47x3x51x4x52x7x54x8x55x13x56)g21+(x1x39x3x44x4x45+x6x47x7x49x8x50x13x53)g14+(x2x40x5x46x9x51x10x52x14x54x15x55x20x56)g13+(x7x32+x13x36+x14x38+x20x43+x21x45+x25x50+x26x52+x28x55)g9+(x1x33x3x37x4x38x7x42x8x43+x12x47x13x48)g7+(x5x39+x6x40x9x44x10x45x14x49x15x50x20x53)g6+(x2x34x11x46x16x51x17x52x21x54x22x55x25x56)g5+(x1x32+x3x36+x5x38+x9x43+x11x45+x16x50+x18x52+x23x55)g3+(x1x31+x4x36+x5x37+x7x41+x10x43+x11x44+x14x48+x17x50+x18x51+x21x53+x24x55+x26x56)g2+(x7x29+x13x31+x14x33+x20x37+x21x39+x25x44+x26x46+x28x51)g1+(x2x30+x6x34+x12x40+x19x4716)h8+(x1x29+x2x30+x3x31+x4x32+x6x34+x7x35+x8x36+x12x40+x13x4130)h7+(x1x29+x2x30+x3x31+x4x32+x5x33+x6x34+x7x35+x8x36+x9x37+x10x38+x13x41+x14x42+x15x43+x20x4844)h6+(x1x29+x2x30+x3x31+x4x32+x5x33+x7x35+x8x36+x9x37+x10x38+x11x39+x13x41+x14x42+x15x43+x16x44+x17x45+x20x48+x21x49+x22x50+x25x5358)h5+(x1x29+x3x31+x4x32+x5x33+x7x35+x8x36+x9x37+x10x38+x11x39+x13x41+x14x42+x15x43+x16x44+x17x45+x18x46+x20x48+x21x49+x22x50+x23x51+x24x52+x25x53+x26x54+x27x55+x28x5672)h4+(x1x29+x3x31+x5x33+x7x35+x9x37+x11x39+x13x41+x14x42+x16x44+x18x46+x20x48+x21x49+x23x51+x25x53+x26x54+x28x5648)h3+(x1x29+x4x32+x5x33+x7x35+x10x38+x11x39+x14x42+x17x45+x18x46+x21x49+x24x52+x26x5436)h2+(x7x35+x13x41+x14x42+x20x48+x21x49+x25x53+x26x54+x28x5624)h1+(x1x35+x3x41+x5x42+x9x48+x11x49+x16x53+x18x54+x23x56)g1+(x3x29+x8x32+x9x33+x13x35+x15x38+x16x39+x20x42+x22x45+x23x46+x25x49+x27x52+x28x54)g2+(x4x29+x8x31+x10x33+x15x37+x17x39+x22x44+x24x46+x27x51)g3+(x6x30x18x39x23x44x24x45x26x49x27x50x28x53)g5+(x11x33+x12x34x16x37x17x38x21x42x22x43x25x48)g6+(x5x29x9x31x10x32x14x35x15x36+x19x40x20x41)g7+(x4x35+x8x41+x10x42+x15x48+x17x49+x22x53+x24x54+x27x56)g9+(x12x30x18x33x23x37x24x38x26x42x27x43x28x48)g13+(x11x29x16x31x17x32+x19x34x21x35x22x36x25x41)g14+(x18x29+x19x30x23x31x24x32x26x35x27x36x28x41)g21[e,f] - h = \left(-x_{1} x_{46} +x_{2} x_{47} -x_{3} x_{51} -x_{4} x_{52} -x_{7} x_{54} -x_{8} x_{55} -x_{13} x_{56} \right)g_{21}+\left(-x_{1} x_{39} -x_{3} x_{44} -x_{4} x_{45} +x_{6} x_{47} -x_{7} x_{49} -x_{8} x_{50} -x_{13} x_{53} \right)g_{14}+\left(x_{2} x_{40} -x_{5} x_{46} -x_{9} x_{51} -x_{10} x_{52} -x_{14} x_{54} -x_{15} x_{55} -x_{20} x_{56} \right)g_{13}+\left(x_{7} x_{32} +x_{13} x_{36} +x_{14} x_{38} +x_{20} x_{43} +x_{21} x_{45} +x_{25} x_{50} +x_{26} x_{52} +x_{28} x_{55} \right)g_{9}+\left(-x_{1} x_{33} -x_{3} x_{37} -x_{4} x_{38} -x_{7} x_{42} -x_{8} x_{43} +x_{12} x_{47} -x_{13} x_{48} \right)g_{7}+\left(-x_{5} x_{39} +x_{6} x_{40} -x_{9} x_{44} -x_{10} x_{45} -x_{14} x_{49} -x_{15} x_{50} -x_{20} x_{53} \right)g_{6}+\left(x_{2} x_{34} -x_{11} x_{46} -x_{16} x_{51} -x_{17} x_{52} -x_{21} x_{54} -x_{22} x_{55} -x_{25} x_{56} \right)g_{5}+\left(x_{1} x_{32} +x_{3} x_{36} +x_{5} x_{38} +x_{9} x_{43} +x_{11} x_{45} +x_{16} x_{50} +x_{18} x_{52} +x_{23} x_{55} \right)g_{3}+\left(x_{1} x_{31} +x_{4} x_{36} +x_{5} x_{37} +x_{7} x_{41} +x_{10} x_{43} +x_{11} x_{44} +x_{14} x_{48} +x_{17} x_{50} +x_{18} x_{51} +x_{21} x_{53} +x_{24} x_{55} +x_{26} x_{56} \right)g_{2}+\left(x_{7} x_{29} +x_{13} x_{31} +x_{14} x_{33} +x_{20} x_{37} +x_{21} x_{39} +x_{25} x_{44} +x_{26} x_{46} +x_{28} x_{51} \right)g_{1}+\left(x_{2} x_{30} +x_{6} x_{34} +x_{12} x_{40} +x_{19} x_{47} -16\right)h_{8}+\left(x_{1} x_{29} +x_{2} x_{30} +x_{3} x_{31} +x_{4} x_{32} +x_{6} x_{34} +x_{7} x_{35} +x_{8} x_{36} +x_{12} x_{40} +x_{13} x_{41} -30\right)h_{7}+\left(x_{1} x_{29} +x_{2} x_{30} +x_{3} x_{31} +x_{4} x_{32} +x_{5} x_{33} +x_{6} x_{34} +x_{7} x_{35} +x_{8} x_{36} +x_{9} x_{37} +x_{10} x_{38} +x_{13} x_{41} +x_{14} x_{42} +x_{15} x_{43} +x_{20} x_{48} -44\right)h_{6}+\left(x_{1} x_{29} +x_{2} x_{30} +x_{3} x_{31} +x_{4} x_{32} +x_{5} x_{33} +x_{7} x_{35} +x_{8} x_{36} +x_{9} x_{37} +x_{10} x_{38} +x_{11} x_{39} +x_{13} x_{41} +x_{14} x_{42} +x_{15} x_{43} +x_{16} x_{44} +x_{17} x_{45} +x_{20} x_{48} +x_{21} x_{49} +x_{22} x_{50} +x_{25} x_{53} -58\right)h_{5}+\left(x_{1} x_{29} +x_{3} x_{31} +x_{4} x_{32} +x_{5} x_{33} +x_{7} x_{35} +x_{8} x_{36} +x_{9} x_{37} +x_{10} x_{38} +x_{11} x_{39} +x_{13} x_{41} +x_{14} x_{42} +x_{15} x_{43} +x_{16} x_{44} +x_{17} x_{45} +x_{18} x_{46} +x_{20} x_{48} +x_{21} x_{49} +x_{22} x_{50} +x_{23} x_{51} +x_{24} x_{52} +x_{25} x_{53} +x_{26} x_{54} +x_{27} x_{55} +x_{28} x_{56} -72\right)h_{4}+\left(x_{1} x_{29} +x_{3} x_{31} +x_{5} x_{33} +x_{7} x_{35} +x_{9} x_{37} +x_{11} x_{39} +x_{13} x_{41} +x_{14} x_{42} +x_{16} x_{44} +x_{18} x_{46} +x_{20} x_{48} +x_{21} x_{49} +x_{23} x_{51} +x_{25} x_{53} +x_{26} x_{54} +x_{28} x_{56} -48\right)h_{3}+\left(x_{1} x_{29} +x_{4} x_{32} +x_{5} x_{33} +x_{7} x_{35} +x_{10} x_{38} +x_{11} x_{39} +x_{14} x_{42} +x_{17} x_{45} +x_{18} x_{46} +x_{21} x_{49} +x_{24} x_{52} +x_{26} x_{54} -36\right)h_{2}+\left(x_{7} x_{35} +x_{13} x_{41} +x_{14} x_{42} +x_{20} x_{48} +x_{21} x_{49} +x_{25} x_{53} +x_{26} x_{54} +x_{28} x_{56} -24\right)h_{1}+\left(x_{1} x_{35} +x_{3} x_{41} +x_{5} x_{42} +x_{9} x_{48} +x_{11} x_{49} +x_{16} x_{53} +x_{18} x_{54} +x_{23} x_{56} \right)g_{-1}+\left(x_{3} x_{29} +x_{8} x_{32} +x_{9} x_{33} +x_{13} x_{35} +x_{15} x_{38} +x_{16} x_{39} +x_{20} x_{42} +x_{22} x_{45} +x_{23} x_{46} +x_{25} x_{49} +x_{27} x_{52} +x_{28} x_{54} \right)g_{-2}+\left(x_{4} x_{29} +x_{8} x_{31} +x_{10} x_{33} +x_{15} x_{37} +x_{17} x_{39} +x_{22} x_{44} +x_{24} x_{46} +x_{27} x_{51} \right)g_{-3}+\left(x_{6} x_{30} -x_{18} x_{39} -x_{23} x_{44} -x_{24} x_{45} -x_{26} x_{49} -x_{27} x_{50} -x_{28} x_{53} \right)g_{-5}+\left(-x_{11} x_{33} +x_{12} x_{34} -x_{16} x_{37} -x_{17} x_{38} -x_{21} x_{42} -x_{22} x_{43} -x_{25} x_{48} \right)g_{-6}+\left(-x_{5} x_{29} -x_{9} x_{31} -x_{10} x_{32} -x_{14} x_{35} -x_{15} x_{36} +x_{19} x_{40} -x_{20} x_{41} \right)g_{-7}+\left(x_{4} x_{35} +x_{8} x_{41} +x_{10} x_{42} +x_{15} x_{48} +x_{17} x_{49} +x_{22} x_{53} +x_{24} x_{54} +x_{27} x_{56} \right)g_{-9}+\left(x_{12} x_{30} -x_{18} x_{33} -x_{23} x_{37} -x_{24} x_{38} -x_{26} x_{42} -x_{27} x_{43} -x_{28} x_{48} \right)g_{-13}+\left(-x_{11} x_{29} -x_{16} x_{31} -x_{17} x_{32} +x_{19} x_{34} -x_{21} x_{35} -x_{22} x_{36} -x_{25} x_{41} \right)g_{-14}+\left(-x_{18} x_{29} +x_{19} x_{30} -x_{23} x_{31} -x_{24} x_{32} -x_{26} x_{35} -x_{27} x_{36} -x_{28} x_{41} \right)g_{-21}
The polynomial system that corresponds to finding the h, e, f triple:
x1x29+x4x32+x5x33+x7x35+x10x38+x11x39+x14x42+x17x45+x18x46+x21x49+x24x52+x26x5436=0x1x29+x3x31+x5x33+x7x35+x9x37+x11x39+x13x41+x14x42+x16x44+x18x46+x20x48+x21x49+x23x51+x25x53+x26x54+x28x5648=0x1x29+x3x31+x4x32+x5x33+x7x35+x8x36+x9x37+x10x38+x11x39+x13x41+x14x42+x15x43+x16x44+x17x45+x18x46+x20x48+x21x49+x22x50+x23x51+x24x52+x25x53+x26x54+x27x55+x28x5672=0x1x29+x2x30+x3x31+x4x32+x5x33+x7x35+x8x36+x9x37+x10x38+x11x39+x13x41+x14x42+x15x43+x16x44+x17x45+x20x48+x21x49+x22x50+x25x5358=0x1x29+x2x30+x3x31+x4x32+x5x33+x6x34+x7x35+x8x36+x9x37+x10x38+x13x41+x14x42+x15x43+x20x4844=0x1x29+x2x30+x3x31+x4x32+x6x34+x7x35+x8x36+x12x40+x13x4130=0x1x31+x4x36+x5x37+x7x41+x10x43+x11x44+x14x48+x17x50+x18x51+x21x53+x24x55+x26x56=0x1x32+x3x36+x5x38+x9x43+x11x45+x16x50+x18x52+x23x55=0x1x33x3x37x4x38x7x42x8x43+x12x47x13x48=0x1x35+x3x41+x5x42+x9x48+x11x49+x16x53+x18x54+x23x56=0x1x39x3x44x4x45+x6x47x7x49x8x50x13x53=0x1x46+x2x47x3x51x4x52x7x54x8x55x13x56=0x2x30+x6x34+x12x40+x19x4716=0x2x34x11x46x16x51x17x52x21x54x22x55x25x56=0x2x40x5x46x9x51x10x52x14x54x15x55x20x56=0x3x29+x8x32+x9x33+x13x35+x15x38+x16x39+x20x42+x22x45+x23x46+x25x49+x27x52+x28x54=0x4x29+x8x31+x10x33+x15x37+x17x39+x22x44+x24x46+x27x51=0x4x35+x8x41+x10x42+x15x48+x17x49+x22x53+x24x54+x27x56=0x5x29x9x31x10x32x14x35x15x36+x19x40x20x41=0x5x39+x6x40x9x44x10x45x14x49x15x50x20x53=0x6x30x18x39x23x44x24x45x26x49x27x50x28x53=0x7x29+x13x31+x14x33+x20x37+x21x39+x25x44+x26x46+x28x51=0x7x32+x13x36+x14x38+x20x43+x21x45+x25x50+x26x52+x28x55=0x7x35+x13x41+x14x42+x20x48+x21x49+x25x53+x26x54+x28x5624=0x11x29x16x31x17x32+x19x34x21x35x22x36x25x41=0x11x33+x12x34x16x37x17x38x21x42x22x43x25x48=0x12x30x18x33x23x37x24x38x26x42x27x43x28x48=0x18x29+x19x30x23x31x24x32x26x35x27x36x28x41=0\begin{array}{rcl}x_{1} x_{29} +x_{4} x_{32} +x_{5} x_{33} +x_{7} x_{35} +x_{10} x_{38} +x_{11} x_{39} +x_{14} x_{42} +x_{17} x_{45} +x_{18} x_{46} +x_{21} x_{49} +x_{24} x_{52} +x_{26} x_{54} -36&=&0\\x_{1} x_{29} +x_{3} x_{31} +x_{5} x_{33} +x_{7} x_{35} +x_{9} x_{37} +x_{11} x_{39} +x_{13} x_{41} +x_{14} x_{42} +x_{16} x_{44} +x_{18} x_{46} +x_{20} x_{48} +x_{21} x_{49} +x_{23} x_{51} +x_{25} x_{53} +x_{26} x_{54} +x_{28} x_{56} -48&=&0\\x_{1} x_{29} +x_{3} x_{31} +x_{4} x_{32} +x_{5} x_{33} +x_{7} x_{35} +x_{8} x_{36} +x_{9} x_{37} +x_{10} x_{38} +x_{11} x_{39} +x_{13} x_{41} +x_{14} x_{42} +x_{15} x_{43} +x_{16} x_{44} +x_{17} x_{45} +x_{18} x_{46} +x_{20} x_{48} +x_{21} x_{49} +x_{22} x_{50} +x_{23} x_{51} +x_{24} x_{52} +x_{25} x_{53} +x_{26} x_{54} +x_{27} x_{55} +x_{28} x_{56} -72&=&0\\x_{1} x_{29} +x_{2} x_{30} +x_{3} x_{31} +x_{4} x_{32} +x_{5} x_{33} +x_{7} x_{35} +x_{8} x_{36} +x_{9} x_{37} +x_{10} x_{38} +x_{11} x_{39} +x_{13} x_{41} +x_{14} x_{42} +x_{15} x_{43} +x_{16} x_{44} +x_{17} x_{45} +x_{20} x_{48} +x_{21} x_{49} +x_{22} x_{50} +x_{25} x_{53} -58&=&0\\x_{1} x_{29} +x_{2} x_{30} +x_{3} x_{31} +x_{4} x_{32} +x_{5} x_{33} +x_{6} x_{34} +x_{7} x_{35} +x_{8} x_{36} +x_{9} x_{37} +x_{10} x_{38} +x_{13} x_{41} +x_{14} x_{42} +x_{15} x_{43} +x_{20} x_{48} -44&=&0\\x_{1} x_{29} +x_{2} x_{30} +x_{3} x_{31} +x_{4} x_{32} +x_{6} x_{34} +x_{7} x_{35} +x_{8} x_{36} +x_{12} x_{40} +x_{13} x_{41} -30&=&0\\x_{1} x_{31} +x_{4} x_{36} +x_{5} x_{37} +x_{7} x_{41} +x_{10} x_{43} +x_{11} x_{44} +x_{14} x_{48} +x_{17} x_{50} +x_{18} x_{51} +x_{21} x_{53} +x_{24} x_{55} +x_{26} x_{56} &=&0\\x_{1} x_{32} +x_{3} x_{36} +x_{5} x_{38} +x_{9} x_{43} +x_{11} x_{45} +x_{16} x_{50} +x_{18} x_{52} +x_{23} x_{55} &=&0\\-x_{1} x_{33} -x_{3} x_{37} -x_{4} x_{38} -x_{7} x_{42} -x_{8} x_{43} +x_{12} x_{47} -x_{13} x_{48} &=&0\\x_{1} x_{35} +x_{3} x_{41} +x_{5} x_{42} +x_{9} x_{48} +x_{11} x_{49} +x_{16} x_{53} +x_{18} x_{54} +x_{23} x_{56} &=&0\\-x_{1} x_{39} -x_{3} x_{44} -x_{4} x_{45} +x_{6} x_{47} -x_{7} x_{49} -x_{8} x_{50} -x_{13} x_{53} &=&0\\-x_{1} x_{46} +x_{2} x_{47} -x_{3} x_{51} -x_{4} x_{52} -x_{7} x_{54} -x_{8} x_{55} -x_{13} x_{56} &=&0\\x_{2} x_{30} +x_{6} x_{34} +x_{12} x_{40} +x_{19} x_{47} -16&=&0\\x_{2} x_{34} -x_{11} x_{46} -x_{16} x_{51} -x_{17} x_{52} -x_{21} x_{54} -x_{22} x_{55} -x_{25} x_{56} &=&0\\x_{2} x_{40} -x_{5} x_{46} -x_{9} x_{51} -x_{10} x_{52} -x_{14} x_{54} -x_{15} x_{55} -x_{20} x_{56} &=&0\\x_{3} x_{29} +x_{8} x_{32} +x_{9} x_{33} +x_{13} x_{35} +x_{15} x_{38} +x_{16} x_{39} +x_{20} x_{42} +x_{22} x_{45} +x_{23} x_{46} +x_{25} x_{49} +x_{27} x_{52} +x_{28} x_{54} &=&0\\x_{4} x_{29} +x_{8} x_{31} +x_{10} x_{33} +x_{15} x_{37} +x_{17} x_{39} +x_{22} x_{44} +x_{24} x_{46} +x_{27} x_{51} &=&0\\x_{4} x_{35} +x_{8} x_{41} +x_{10} x_{42} +x_{15} x_{48} +x_{17} x_{49} +x_{22} x_{53} +x_{24} x_{54} +x_{27} x_{56} &=&0\\-x_{5} x_{29} -x_{9} x_{31} -x_{10} x_{32} -x_{14} x_{35} -x_{15} x_{36} +x_{19} x_{40} -x_{20} x_{41} &=&0\\-x_{5} x_{39} +x_{6} x_{40} -x_{9} x_{44} -x_{10} x_{45} -x_{14} x_{49} -x_{15} x_{50} -x_{20} x_{53} &=&0\\x_{6} x_{30} -x_{18} x_{39} -x_{23} x_{44} -x_{24} x_{45} -x_{26} x_{49} -x_{27} x_{50} -x_{28} x_{53} &=&0\\x_{7} x_{29} +x_{13} x_{31} +x_{14} x_{33} +x_{20} x_{37} +x_{21} x_{39} +x_{25} x_{44} +x_{26} x_{46} +x_{28} x_{51} &=&0\\x_{7} x_{32} +x_{13} x_{36} +x_{14} x_{38} +x_{20} x_{43} +x_{21} x_{45} +x_{25} x_{50} +x_{26} x_{52} +x_{28} x_{55} &=&0\\x_{7} x_{35} +x_{13} x_{41} +x_{14} x_{42} +x_{20} x_{48} +x_{21} x_{49} +x_{25} x_{53} +x_{26} x_{54} +x_{28} x_{56} -24&=&0\\-x_{11} x_{29} -x_{16} x_{31} -x_{17} x_{32} +x_{19} x_{34} -x_{21} x_{35} -x_{22} x_{36} -x_{25} x_{41} &=&0\\-x_{11} x_{33} +x_{12} x_{34} -x_{16} x_{37} -x_{17} x_{38} -x_{21} x_{42} -x_{22} x_{43} -x_{25} x_{48} &=&0\\x_{12} x_{30} -x_{18} x_{33} -x_{23} x_{37} -x_{24} x_{38} -x_{26} x_{42} -x_{27} x_{43} -x_{28} x_{48} &=&0\\-x_{18} x_{29} +x_{19} x_{30} -x_{23} x_{31} -x_{24} x_{32} -x_{26} x_{35} -x_{27} x_{36} -x_{28} x_{41} &=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=16h8+30h7+44h6+58h5+72h4+48h3+36h2+24h1e=x7g47+x1g41+x13g40+x14g38+x3g35+x4g34+x5g33+x20g32+x21g30+x2g29+x8g28+x9g27+x10g26+x11g25+x25g24+x26g23+x6g22+x15g20+x16g19+x17g18+x18g17+x28g16+x12g15+x22g12+x23g11+x24g10+x19g8+x27g4f=g4+g10+g11+g12+g16+g22+g23+g24+g28+g29+g30+g32+g33+g34+g35+g41+g47\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+58h_{5}+72h_{4}+48h_{3}+36h_{2}+24h_{1}\\e&=&x_{7} g_{47}+x_{1} g_{41}+x_{13} g_{40}+x_{14} g_{38}+x_{3} g_{35}+x_{4} g_{34}+x_{5} g_{33}+x_{20} g_{32}+x_{21} g_{30}+x_{2} g_{29}+x_{8} g_{28}+x_{9} g_{27}+x_{10} g_{26}+x_{11} g_{25}+x_{25} g_{24}+x_{26} g_{23}+x_{6} g_{22}+x_{15} g_{20}+x_{16} g_{19}+x_{17} g_{18}+x_{18} g_{17}+x_{28} g_{16}+x_{12} g_{15}+x_{22} g_{12}+x_{23} g_{11}+x_{24} g_{10}+x_{19} g_{8}+x_{27} g_{4}\\f&=&g_{-4}+g_{-10}+g_{-11}+g_{-12}+g_{-16}+g_{-22}+g_{-23}+g_{-24}+g_{-28}+g_{-29}+g_{-30}+g_{-32}+g_{-33}+g_{-34}+g_{-35}+g_{-41}+g_{-47}\end{array}
Matrix form of the system we are trying to solve: (1001101000000000000010010100101010100000000000011010110110111011000000000001111111111111101100000000000111001000111111110000000000010000000011110111000000000000000000001001000000000100110010010100101000000000000101000010000010000000000010000000000000001000000010100001010000100000010001000000000000000000000001000000000000011000110010000010000110001000000000001011001100110000100000000000000000010001010000000000000000100001000000000010100001010010000010001100011000000000000000000100000000000000000001110000001000001100000000000001000000100000100000000000110100000010000000000001100011010000001100001000000000000000000000001100011000010000000000000000001000011010110000000000000000110000000000001000000000000001000001000000000100000000000001100001000000000000000000000000011000110110)[col. vect.]=(364872584430000016000000000240000000)\begin{pmatrix}1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1\\ 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1\\ 0 & 0 & -1 & -1 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 1 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & -1 & -1 & 0 & -1 & -1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}36\\ 48\\ 72\\ 58\\ 44\\ 30\\ 0\\ 0\\ 0\\ 0\\ 16\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 24\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=16h8+30h7+44h6+58h5+72h4+48h3+36h2+24h1e=x7g47+x1g41+x13g40+x14g38+x3g35+x4g34+x5g33+x20g32+x21g30+x2g29+x8g28+x9g27+x10g26+x11g25+x25g24+x26g23+x6g22+x15g20+x16g19+x17g18+x18g17+x28g16+x12g15+x22g12+x23g11+x24g10+x19g8+x27g4f=x55g4+x47g8+x52g10+x51g11+x50g12+x40g15+x56g16+x46g17+x45g18+x44g19+x43g20+x34g22+x54g23+x53g24+x39g25+x38g26+x37g27+x36g28+x30g29+x49g30+x48g32+x33g33+x32g34+x31g35+x42g38+x41g40+x29g41+x35g47\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+58h_{5}+72h_{4}+48h_{3}+36h_{2}+24h_{1}\\ e&=&x_{7} g_{47}+x_{1} g_{41}+x_{13} g_{40}+x_{14} g_{38}+x_{3} g_{35}+x_{4} g_{34}+x_{5} g_{33}+x_{20} g_{32}+x_{21} g_{30}+x_{2} g_{29}+x_{8} g_{28}+x_{9} g_{27}+x_{10} g_{26}+x_{11} g_{25}+x_{25} g_{24}+x_{26} g_{23}+x_{6} g_{22}+x_{15} g_{20}+x_{16} g_{19}+x_{17} g_{18}+x_{18} g_{17}+x_{28} g_{16}+x_{12} g_{15}+x_{22} g_{12}+x_{23} g_{11}+x_{24} g_{10}+x_{19} g_{8}+x_{27} g_{4}\\ f&=&x_{55} g_{-4}+x_{47} g_{-8}+x_{52} g_{-10}+x_{51} g_{-11}+x_{50} g_{-12}+x_{40} g_{-15}+x_{56} g_{-16}+x_{46} g_{-17}+x_{45} g_{-18}+x_{44} g_{-19}+x_{43} g_{-20}+x_{34} g_{-22}+x_{54} g_{-23}+x_{53} g_{-24}+x_{39} g_{-25}+x_{38} g_{-26}+x_{37} g_{-27}+x_{36} g_{-28}+x_{30} g_{-29}+x_{49} g_{-30}+x_{48} g_{-32}+x_{33} g_{-33}+x_{32} g_{-34}+x_{31} g_{-35}+x_{42} g_{-38}+x_{41} g_{-40}+x_{29} g_{-41}+x_{35} g_{-47}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x29+x4x32+x5x33+x7x35+x10x38+x11x39+x14x42+x17x45+x18x46+x21x49+x24x52+x26x5436=0x1x29+x3x31+x5x33+x7x35+x9x37+x11x39+x13x41+x14x42+x16x44+x18x46+x20x48+x21x49+x23x51+x25x53+x26x54+x28x5648=0x1x29+x3x31+x4x32+x5x33+x7x35+x8x36+x9x37+x10x38+x11x39+x13x41+x14x42+x15x43+x16x44+x17x45+x18x46+x20x48+x21x49+x22x50+x23x51+x24x52+x25x53+x26x54+x27x55+x28x5672=0x1x29+x2x30+x3x31+x4x32+x5x33+x7x35+x8x36+x9x37+x10x38+x11x39+x13x41+x14x42+x15x43+x16x44+x17x45+x20x48+x21x49+x22x50+x25x5358=0x1x29+x2x30+x3x31+x4x32+x5x33+x6x34+x7x35+x8x36+x9x37+x10x38+x13x41+x14x42+x15x43+x20x4844=0x1x29+x2x30+x3x31+x4x32+x6x34+x7x35+x8x36+x12x40+x13x4130=0x1x31+x4x36+x5x37+x7x41+x10x43+x11x44+x14x48+x17x50+x18x51+x21x53+x24x55+x26x56=0x1x32+x3x36+x5x38+x9x43+x11x45+x16x50+x18x52+x23x55=0x1x33x3x37x4x38x7x42x8x43+x12x47x13x48=0x1x35+x3x41+x5x42+x9x48+x11x49+x16x53+x18x54+x23x56=0x1x39x3x44x4x45+x6x47x7x49x8x50x13x53=0x1x46+x2x47x3x51x4x52x7x54x8x55x13x56=0x2x30+x6x34+x12x40+x19x4716=0x2x34x11x46x16x51x17x52x21x54x22x55x25x56=0x2x40x5x46x9x51x10x52x14x54x15x55x20x56=0x3x29+x8x32+x9x33+x13x35+x15x38+x16x39+x20x42+x22x45+x23x46+x25x49+x27x52+x28x54=0x4x29+x8x31+x10x33+x15x37+x17x39+x22x44+x24x46+x27x51=0x4x35+x8x41+x10x42+x15x48+x17x49+x22x53+x24x54+x27x56=0x5x29x9x31x10x32x14x35x15x36+x19x40x20x41=0x5x39+x6x40x9x44x10x45x14x49x15x50x20x53=0x6x30x18x39x23x44x24x45x26x49x27x50x28x53=0x7x29+x13x31+x14x33+x20x37+x21x39+x25x44+x26x46+x28x51=0x7x32+x13x36+x14x38+x20x43+x21x45+x25x50+x26x52+x28x55=0x7x35+x13x41+x14x42+x20x48+x21x49+x25x53+x26x54+x28x5624=0x11x29x16x31x17x32+x19x34x21x35x22x36x25x41=0x11x33+x12x34x16x37x17x38x21x42x22x43x25x48=0x12x30x18x33x23x37x24x38x26x42x27x43x28x48=0x18x29+x19x30x23x31x24x32x26x35x27x36x28x41=0\begin{array}{rcl}x_{1} x_{29} +x_{4} x_{32} +x_{5} x_{33} +x_{7} x_{35} +x_{10} x_{38} +x_{11} x_{39} +x_{14} x_{42} +x_{17} x_{45} +x_{18} x_{46} +x_{21} x_{49} +x_{24} x_{52} +x_{26} x_{54} -36&=&0\\x_{1} x_{29} +x_{3} x_{31} +x_{5} x_{33} +x_{7} x_{35} +x_{9} x_{37} +x_{11} x_{39} +x_{13} x_{41} +x_{14} x_{42} +x_{16} x_{44} +x_{18} x_{46} +x_{20} x_{48} +x_{21} x_{49} +x_{23} x_{51} +x_{25} x_{53} +x_{26} x_{54} +x_{28} x_{56} -48&=&0\\x_{1} x_{29} +x_{3} x_{31} +x_{4} x_{32} +x_{5} x_{33} +x_{7} x_{35} +x_{8} x_{36} +x_{9} x_{37} +x_{10} x_{38} +x_{11} x_{39} +x_{13} x_{41} +x_{14} x_{42} +x_{15} x_{43} +x_{16} x_{44} +x_{17} x_{45} +x_{18} x_{46} +x_{20} x_{48} +x_{21} x_{49} +x_{22} x_{50} +x_{23} x_{51} +x_{24} x_{52} +x_{25} x_{53} +x_{26} x_{54} +x_{27} x_{55} +x_{28} x_{56} -72&=&0\\x_{1} x_{29} +x_{2} x_{30} +x_{3} x_{31} +x_{4} x_{32} +x_{5} x_{33} +x_{7} x_{35} +x_{8} x_{36} +x_{9} x_{37} +x_{10} x_{38} +x_{11} x_{39} +x_{13} x_{41} +x_{14} x_{42} +x_{15} x_{43} +x_{16} x_{44} +x_{17} x_{45} +x_{20} x_{48} +x_{21} x_{49} +x_{22} x_{50} +x_{25} x_{53} -58&=&0\\x_{1} x_{29} +x_{2} x_{30} +x_{3} x_{31} +x_{4} x_{32} +x_{5} x_{33} +x_{6} x_{34} +x_{7} x_{35} +x_{8} x_{36} +x_{9} x_{37} +x_{10} x_{38} +x_{13} x_{41} +x_{14} x_{42} +x_{15} x_{43} +x_{20} x_{48} -44&=&0\\x_{1} x_{29} +x_{2} x_{30} +x_{3} x_{31} +x_{4} x_{32} +x_{6} x_{34} +x_{7} x_{35} +x_{8} x_{36} +x_{12} x_{40} +x_{13} x_{41} -30&=&0\\x_{1} x_{31} +x_{4} x_{36} +x_{5} x_{37} +x_{7} x_{41} +x_{10} x_{43} +x_{11} x_{44} +x_{14} x_{48} +x_{17} x_{50} +x_{18} x_{51} +x_{21} x_{53} +x_{24} x_{55} +x_{26} x_{56} &=&0\\x_{1} x_{32} +x_{3} x_{36} +x_{5} x_{38} +x_{9} x_{43} +x_{11} x_{45} +x_{16} x_{50} +x_{18} x_{52} +x_{23} x_{55} &=&0\\-x_{1} x_{33} -x_{3} x_{37} -x_{4} x_{38} -x_{7} x_{42} -x_{8} x_{43} +x_{12} x_{47} -x_{13} x_{48} &=&0\\x_{1} x_{35} +x_{3} x_{41} +x_{5} x_{42} +x_{9} x_{48} +x_{11} x_{49} +x_{16} x_{53} +x_{18} x_{54} +x_{23} x_{56} &=&0\\-x_{1} x_{39} -x_{3} x_{44} -x_{4} x_{45} +x_{6} x_{47} -x_{7} x_{49} -x_{8} x_{50} -x_{13} x_{53} &=&0\\-x_{1} x_{46} +x_{2} x_{47} -x_{3} x_{51} -x_{4} x_{52} -x_{7} x_{54} -x_{8} x_{55} -x_{13} x_{56} &=&0\\x_{2} x_{30} +x_{6} x_{34} +x_{12} x_{40} +x_{19} x_{47} -16&=&0\\x_{2} x_{34} -x_{11} x_{46} -x_{16} x_{51} -x_{17} x_{52} -x_{21} x_{54} -x_{22} x_{55} -x_{25} x_{56} &=&0\\x_{2} x_{40} -x_{5} x_{46} -x_{9} x_{51} -x_{10} x_{52} -x_{14} x_{54} -x_{15} x_{55} -x_{20} x_{56} &=&0\\x_{3} x_{29} +x_{8} x_{32} +x_{9} x_{33} +x_{13} x_{35} +x_{15} x_{38} +x_{16} x_{39} +x_{20} x_{42} +x_{22} x_{45} +x_{23} x_{46} +x_{25} x_{49} +x_{27} x_{52} +x_{28} x_{54} &=&0\\x_{4} x_{29} +x_{8} x_{31} +x_{10} x_{33} +x_{15} x_{37} +x_{17} x_{39} +x_{22} x_{44} +x_{24} x_{46} +x_{27} x_{51} &=&0\\x_{4} x_{35} +x_{8} x_{41} +x_{10} x_{42} +x_{15} x_{48} +x_{17} x_{49} +x_{22} x_{53} +x_{24} x_{54} +x_{27} x_{56} &=&0\\-x_{5} x_{29} -x_{9} x_{31} -x_{10} x_{32} -x_{14} x_{35} -x_{15} x_{36} +x_{19} x_{40} -x_{20} x_{41} &=&0\\-x_{5} x_{39} +x_{6} x_{40} -x_{9} x_{44} -x_{10} x_{45} -x_{14} x_{49} -x_{15} x_{50} -x_{20} x_{53} &=&0\\x_{6} x_{30} -x_{18} x_{39} -x_{23} x_{44} -x_{24} x_{45} -x_{26} x_{49} -x_{27} x_{50} -x_{28} x_{53} &=&0\\x_{7} x_{29} +x_{13} x_{31} +x_{14} x_{33} +x_{20} x_{37} +x_{21} x_{39} +x_{25} x_{44} +x_{26} x_{46} +x_{28} x_{51} &=&0\\x_{7} x_{32} +x_{13} x_{36} +x_{14} x_{38} +x_{20} x_{43} +x_{21} x_{45} +x_{25} x_{50} +x_{26} x_{52} +x_{28} x_{55} &=&0\\x_{7} x_{35} +x_{13} x_{41} +x_{14} x_{42} +x_{20} x_{48} +x_{21} x_{49} +x_{25} x_{53} +x_{26} x_{54} +x_{28} x_{56} -24&=&0\\-x_{11} x_{29} -x_{16} x_{31} -x_{17} x_{32} +x_{19} x_{34} -x_{21} x_{35} -x_{22} x_{36} -x_{25} x_{41} &=&0\\-x_{11} x_{33} +x_{12} x_{34} -x_{16} x_{37} -x_{17} x_{38} -x_{21} x_{42} -x_{22} x_{43} -x_{25} x_{48} &=&0\\x_{12} x_{30} -x_{18} x_{33} -x_{23} x_{37} -x_{24} x_{38} -x_{26} x_{42} -x_{27} x_{43} -x_{28} x_{48} &=&0\\-x_{18} x_{29} +x_{19} x_{30} -x_{23} x_{31} -x_{24} x_{32} -x_{26} x_{35} -x_{27} x_{36} -x_{28} x_{41} &=&0\\\end{array}

A185A^{85}_1
h-characteristic: (1, 0, 0, 1, 0, 1, 0, 2)
Length of the weight dual to h: 170
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A7+A1A^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 2: E6+A1E^{1}_6+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V16ψ+V14ψ+2V13ψ+2V12ψ+2V11ψ+2V10ψ+2V9ψ+3V8ψ+2V7ψ+V6ψ+2V5ψ+3V4ψ+2V3ψ+2V2ψ+2Vψ+V0V_{16\psi}+V_{14\psi}+2V_{13\psi}+2V_{12\psi}+2V_{11\psi}+2V_{10\psi}+2V_{9\psi}+3V_{8\psi}+2V_{7\psi}+V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=16h8+30h7+44h6+57h5+70h4+47h3+35h2+24h1e=7g41+g31+7g28+15g27+15g26+12g24+12g23+16g15f=g15+g23+g24+g26+g27+g28+g31+g41\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\\ e&=&7g_{41}+g_{31}+7g_{28}+15g_{27}+15g_{26}+12g_{24}+12g_{23}+16g_{15}\\ f&=&g_{-15}+g_{-23}+g_{-24}+g_{-26}+g_{-27}+g_{-28}+g_{-31}+g_{-41}\end{array}
Lie brackets of the above elements.
h=16h8+30h7+44h6+57h5+70h4+47h3+35h2+24h1e=7g41+g31+7g28+15g27+15g26+12g24+12g23+16g15f=g15+g23+g24+g26+g27+g28+g31+g41\begin{array}{rcl}[e, f]&=&16h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\\ [h, e]&=&14g_{41}+2g_{31}+14g_{28}+30g_{27}+30g_{26}+24g_{24}+24g_{23}+32g_{15}\\ [h, f]&=&-2g_{-15}-2g_{-23}-2g_{-24}-2g_{-26}-2g_{-27}-2g_{-28}-2g_{-31}-2g_{-41}\end{array}
Centralizer type: 00
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Unknown elements.
h=16h8+30h7+44h6+57h5+70h4+47h3+35h2+24h1e=x1g41+x8g31+x7g28+x5g27+x3g26+x2g24+x6g23+x4g15e=x12g15+x14g23+x10g24+x11g26+x13g27+x15g28+x16g31+x9g41\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\\ e&=&x_{1} g_{41}+x_{8} g_{31}+x_{7} g_{28}+x_{5} g_{27}+x_{3} g_{26}+x_{2} g_{24}+x_{6} g_{23}+x_{4} g_{15}\\ f&=&x_{12} g_{-15}+x_{14} g_{-23}+x_{10} g_{-24}+x_{11} g_{-26}+x_{13} g_{-27}+x_{15} g_{-28}+x_{16} g_{-31}+x_{9} g_{-41}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x4x1216)h8+(x1x9+x4x12+x7x1530)h7+(x1x9+x3x11+x5x13+x7x1544)h6+(x1x9+x2x10+x3x11+x5x13+x7x15+x8x1657)h5+(x1x9+x2x10+x3x11+x5x13+x6x14+x7x15+2x8x1670)h4+(x1x9+x2x10+x5x13+x6x14+x8x1647)h3+(x1x9+x3x11+x6x14+x8x1635)h2+(x2x10+x6x1424)h1[e,f] - h = \left(x_{4} x_{12} -16\right)h_{8}+\left(x_{1} x_{9} +x_{4} x_{12} +x_{7} x_{15} -30\right)h_{7}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} -44\right)h_{6}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -57\right)h_{5}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +2x_{8} x_{16} -70\right)h_{4}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{6} x_{14} +x_{8} x_{16} -47\right)h_{3}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{8} x_{16} -35\right)h_{2}+\left(x_{2} x_{10} +x_{6} x_{14} -24\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x9+x3x11+x6x14+x8x1635=0x1x9+x2x10+x5x13+x6x14+x8x1647=0x1x9+x2x10+x3x11+x5x13+x6x14+x7x15+2x8x1670=0x1x9+x2x10+x3x11+x5x13+x7x15+x8x1657=0x1x9+x3x11+x5x13+x7x1544=0x1x9+x4x12+x7x1530=0x2x10+x6x1424=0x4x1216=0\begin{array}{rcl}x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{8} x_{16} -35&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{6} x_{14} +x_{8} x_{16} -47&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +2x_{8} x_{16} -70&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -57&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} -44&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{7} x_{15} -30&=&0\\x_{2} x_{10} +x_{6} x_{14} -24&=&0\\x_{4} x_{12} -16&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=16h8+30h7+44h6+57h5+70h4+47h3+35h2+24h1e=x1g41+x8g31+x7g28+x5g27+x3g26+x2g24+x6g23+x4g15f=g15+g23+g24+g26+g27+g28+g31+g41\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\\e&=&x_{1} g_{41}+x_{8} g_{31}+x_{7} g_{28}+x_{5} g_{27}+x_{3} g_{26}+x_{2} g_{24}+x_{6} g_{23}+x_{4} g_{15}\\f&=&g_{-15}+g_{-23}+g_{-24}+g_{-26}+g_{-27}+g_{-28}+g_{-31}+g_{-41}\end{array}
Matrix form of the system we are trying to solve: (1010010111001101111011121110101110101010100100100100010000010000)[col. vect.]=(3547705744302416)\begin{pmatrix}1 & 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1\\ 1 & 1 & 1 & 0 & 1 & 1 & 1 & 2\\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1\\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0\\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}35\\ 47\\ 70\\ 57\\ 44\\ 30\\ 24\\ 16\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=16h8+30h7+44h6+57h5+70h4+47h3+35h2+24h1e=x1g41+x8g31+x7g28+x5g27+x3g26+x2g24+x6g23+x4g15f=x12g15+x14g23+x10g24+x11g26+x13g27+x15g28+x16g31+x9g41\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\\ e&=&x_{1} g_{41}+x_{8} g_{31}+x_{7} g_{28}+x_{5} g_{27}+x_{3} g_{26}+x_{2} g_{24}+x_{6} g_{23}+x_{4} g_{15}\\ f&=&x_{12} g_{-15}+x_{14} g_{-23}+x_{10} g_{-24}+x_{11} g_{-26}+x_{13} g_{-27}+x_{15} g_{-28}+x_{16} g_{-31}+x_{9} g_{-41}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x9+x3x11+x6x14+x8x1635=0x1x9+x2x10+x5x13+x6x14+x8x1647=0x1x9+x2x10+x3x11+x5x13+x6x14+x7x15+2x8x1670=0x1x9+x2x10+x3x11+x5x13+x7x15+x8x1657=0x1x9+x3x11+x5x13+x7x1544=0x1x9+x4x12+x7x1530=0x2x10+x6x1424=0x4x1216=0\begin{array}{rcl}x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{8} x_{16} -35&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{6} x_{14} +x_{8} x_{16} -47&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +2x_{8} x_{16} -70&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -57&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} -44&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{7} x_{15} -30&=&0\\x_{2} x_{10} +x_{6} x_{14} -24&=&0\\x_{4} x_{12} -16&=&0\\\end{array}

A184A^{84}_1
h-characteristic: (2, 0, 0, 0, 0, 2, 0, 2)
Length of the weight dual to h: 168
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A7A^{1}_7 Containing regular semisimple subalgebra number 2: E6E^{1}_6
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V16ψ+V14ψ+6V12ψ+2V10ψ+7V8ψ+V6ψ+7V4ψ+V2ψ+8V0V_{16\psi}+V_{14\psi}+6V_{12\psi}+2V_{10\psi}+7V_{8\psi}+V_{6\psi}+7V_{4\psi}+V_{2\psi}+8V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=16h8+30h7+44h6+56h5+68h4+46h3+34h2+24h1e=7g54+15g27+15g26+12g24+12g23+16g15+7g14f=g14+g15+g23+g24+g26+g27+g54\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+56h_{5}+68h_{4}+46h_{3}+34h_{2}+24h_{1}\\ e&=&7g_{54}+15g_{27}+15g_{26}+12g_{24}+12g_{23}+16g_{15}+7g_{14}\\ f&=&g_{-14}+g_{-15}+g_{-23}+g_{-24}+g_{-26}+g_{-27}+g_{-54}\end{array}
Lie brackets of the above elements.
h=16h8+30h7+44h6+56h5+68h4+46h3+34h2+24h1e=7g54+15g27+15g26+12g24+12g23+16g15+7g14f=g14+g15+g23+g24+g26+g27+g54\begin{array}{rcl}[e, f]&=&16h_{8}+30h_{7}+44h_{6}+56h_{5}+68h_{4}+46h_{3}+34h_{2}+24h_{1}\\ [h, e]&=&14g_{54}+30g_{27}+30g_{26}+24g_{24}+24g_{23}+32g_{15}+14g_{14}\\ [h, f]&=&-2g_{-14}-2g_{-15}-2g_{-23}-2g_{-24}-2g_{-26}-2g_{-27}-2g_{-54}\end{array}
Centralizer type: A2A_2
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Unknown elements.
h=16h8+30h7+44h6+56h5+68h4+46h3+34h2+24h1e=x1g54+x3g27+x5g26+x6g24+x2g23+x4g15+x7g14e=x14g14+x11g15+x9g23+x13g24+x12g26+x10g27+x8g54\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+56h_{5}+68h_{4}+46h_{3}+34h_{2}+24h_{1}\\ e&=&x_{1} g_{54}+x_{3} g_{27}+x_{5} g_{26}+x_{6} g_{24}+x_{2} g_{23}+x_{4} g_{15}+x_{7} g_{14}\\ f&=&x_{14} g_{-14}+x_{11} g_{-15}+x_{9} g_{-23}+x_{13} g_{-24}+x_{12} g_{-26}+x_{10} g_{-27}+x_{8} g_{-54}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x4x1116)h8+(x1x8+x4x11+x7x1430)h7+(x1x8+x3x10+x5x12+x7x1444)h6+(2x1x8+x3x10+x5x12+x6x1356)h5+(2x1x8+x2x9+x3x10+x5x12+x6x1368)h4+(x1x8+x2x9+x3x10+x6x1346)h3+(x1x8+x2x9+x5x1234)h2+(x2x9+x6x1324)h1[e,f] - h = \left(x_{4} x_{11} -16\right)h_{8}+\left(x_{1} x_{8} +x_{4} x_{11} +x_{7} x_{14} -30\right)h_{7}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -44\right)h_{6}+\left(2x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -56\right)h_{5}+\left(2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -68\right)h_{4}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{6} x_{13} -46\right)h_{3}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{5} x_{12} -34\right)h_{2}+\left(x_{2} x_{9} +x_{6} x_{13} -24\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x2x9+x5x1234=0x1x8+x2x9+x3x10+x6x1346=02x1x8+x2x9+x3x10+x5x12+x6x1368=02x1x8+x3x10+x5x12+x6x1356=0x1x8+x3x10+x5x12+x7x1444=0x1x8+x4x11+x7x1430=0x2x9+x6x1324=0x4x1116=0\begin{array}{rcl}x_{1} x_{8} +x_{2} x_{9} +x_{5} x_{12} -34&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{6} x_{13} -46&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -68&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -56&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -44&=&0\\x_{1} x_{8} +x_{4} x_{11} +x_{7} x_{14} -30&=&0\\x_{2} x_{9} +x_{6} x_{13} -24&=&0\\x_{4} x_{11} -16&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=16h8+30h7+44h6+56h5+68h4+46h3+34h2+24h1e=x1g54+x3g27+x5g26+x6g24+x2g23+x4g15+x7g14f=g14+g15+g23+g24+g26+g27+g54\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+56h_{5}+68h_{4}+46h_{3}+34h_{2}+24h_{1}\\e&=&x_{1} g_{54}+x_{3} g_{27}+x_{5} g_{26}+x_{6} g_{24}+x_{2} g_{23}+x_{4} g_{15}+x_{7} g_{14}\\f&=&g_{-14}+g_{-15}+g_{-23}+g_{-24}+g_{-26}+g_{-27}+g_{-54}\end{array}
Matrix form of the system we are trying to solve: (11001001110010211011020101101010101100100101000100001000)[col. vect.]=(3446685644302416)\begin{pmatrix}1 & 1 & 0 & 0 & 1 & 0 & 0\\ 1 & 1 & 1 & 0 & 0 & 1 & 0\\ 2 & 1 & 1 & 0 & 1 & 1 & 0\\ 2 & 0 & 1 & 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 1 & 0 & 0 & 1 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}34\\ 46\\ 68\\ 56\\ 44\\ 30\\ 24\\ 16\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=16h8+30h7+44h6+56h5+68h4+46h3+34h2+24h1e=x1g54+x3g27+x5g26+x6g24+x2g23+x4g15+x7g14f=x14g14+x11g15+x9g23+x13g24+x12g26+x10g27+x8g54\begin{array}{rcl}h&=&16h_{8}+30h_{7}+44h_{6}+56h_{5}+68h_{4}+46h_{3}+34h_{2}+24h_{1}\\ e&=&x_{1} g_{54}+x_{3} g_{27}+x_{5} g_{26}+x_{6} g_{24}+x_{2} g_{23}+x_{4} g_{15}+x_{7} g_{14}\\ f&=&x_{14} g_{-14}+x_{11} g_{-15}+x_{9} g_{-23}+x_{13} g_{-24}+x_{12} g_{-26}+x_{10} g_{-27}+x_{8} g_{-54}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x2x9+x5x1234=0x1x8+x2x9+x3x10+x6x1346=02x1x8+x2x9+x3x10+x5x12+x6x1368=02x1x8+x3x10+x5x12+x6x1356=0x1x8+x3x10+x5x12+x7x1444=0x1x8+x4x11+x7x1430=0x2x9+x6x1324=0x4x1116=0\begin{array}{rcl}x_{1} x_{8} +x_{2} x_{9} +x_{5} x_{12} -34&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{6} x_{13} -46&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -68&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -56&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -44&=&0\\x_{1} x_{8} +x_{4} x_{11} +x_{7} x_{14} -30&=&0\\x_{2} x_{9} +x_{6} x_{13} -24&=&0\\x_{4} x_{11} -16&=&0\\\end{array}

A184A^{84}_1
h-characteristic: (1, 0, 0, 1, 0, 1, 1, 0)
Length of the weight dual to h: 168
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A7A^{1}_7
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 2V15ψ+V14ψ+3V12ψ+2V11ψ+V10ψ+2V9ψ+3V8ψ+4V7ψ+V6ψ+2V5ψ+3V4ψ+2V3ψ+V2ψ+3V02V_{15\psi}+V_{14\psi}+3V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+V_{2\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=15h8+30h7+44h6+57h5+70h4+47h3+35h2+24h1e=7g33+16g31+12g24+12g23+15g22+15g21+7g20f=g20+g21+g22+g23+g24+g31+g33\begin{array}{rcl}h&=&15h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\\ e&=&7g_{33}+16g_{31}+12g_{24}+12g_{23}+15g_{22}+15g_{21}+7g_{20}\\ f&=&g_{-20}+g_{-21}+g_{-22}+g_{-23}+g_{-24}+g_{-31}+g_{-33}\end{array}
Lie brackets of the above elements.
h=15h8+30h7+44h6+57h5+70h4+47h3+35h2+24h1e=7g33+16g31+12g24+12g23+15g22+15g21+7g20f=g20+g21+g22+g23+g24+g31+g33\begin{array}{rcl}[e, f]&=&15h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\\ [h, e]&=&14g_{33}+32g_{31}+24g_{24}+24g_{23}+30g_{22}+30g_{21}+14g_{20}\\ [h, f]&=&-2g_{-20}-2g_{-21}-2g_{-22}-2g_{-23}-2g_{-24}-2g_{-31}-2g_{-33}\end{array}
Centralizer type: A14A^{4}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 3): h8+h5h3+h2h_{8}+h_{5}-h_{3}+h_{2}, g3g2g5g8g_{3}-g_{-2}-g_{-5}-g_{-8}, g8+g5+g2g3g_{8}+g_{5}+g_{2}-g_{-3}
Basis of centralizer intersected with cartan (dimension: 1): h8h5+h3h2-h_{8}-h_{5}+h_{3}-h_{2}
Cartan of centralizer (dimension: 1): h8h5+h3h2-h_{8}-h_{5}+h_{3}-h_{2}
Cartan-generating semisimple element: h8h5+h3h2-h_{8}-h_{5}+h_{3}-h_{2}
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & -2\\ \end{pmatrix}
Characteristic polynomial ad H: x34xx^3-4x
Factorization of characteristic polynomial of ad H: (x )(x -2)(x +2)
Eigenvalues of ad H: 00, 22, 2-2
3 eigenvectors of ad H: 1, 0, 0(1,0,0), 0, 1, 0(0,1,0), 0, 0, 1(0,0,1)
Centralizer type: A^{4}_1
Reductive components (1 total):
Scalar product computed: (1120)\begin{pmatrix}1/120\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h8h5+h3h2-h_{8}-h_{5}+h_{3}-h_{2}
matching e: g3g2g5g8g_{3}-g_{-2}-g_{-5}-g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & -2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h8h5+h3h2-h_{8}-h_{5}+h_{3}-h_{2}
matching e: g3g2g5g8g_{3}-g_{-2}-g_{-5}-g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & -2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (480)\begin{pmatrix}480\\ \end{pmatrix}
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Unknown elements.
h=15h8+30h7+44h6+57h5+70h4+47h3+35h2+24h1e=x1g33+x4g31+x2g24+x6g23+x3g22+x5g21+x7g20e=x14g20+x12g21+x10g22+x13g23+x9g24+x11g31+x8g33\begin{array}{rcl}h&=&15h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\\ e&=&x_{1} g_{33}+x_{4} g_{31}+x_{2} g_{24}+x_{6} g_{23}+x_{3} g_{22}+x_{5} g_{21}+x_{7} g_{20}\\ f&=&x_{14} g_{-20}+x_{12} g_{-21}+x_{10} g_{-22}+x_{13} g_{-23}+x_{9} g_{-24}+x_{11} g_{-31}+x_{8} g_{-33}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x3x1015)h8+(x3x10+x5x1230)h7+(x1x8+x3x10+x5x12+x7x1444)h6+(x1x8+x2x9+x4x11+x5x12+x7x1457)h5+(x1x8+x2x9+2x4x11+x6x13+x7x1470)h4+(x1x8+x2x9+x4x11+x6x1347)h3+(x1x8+x4x11+x6x1335)h2+(x2x9+x6x1324)h1[e,f] - h = \left(x_{3} x_{10} -15\right)h_{8}+\left(x_{3} x_{10} +x_{5} x_{12} -30\right)h_{7}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -44\right)h_{6}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -57\right)h_{5}+\left(x_{1} x_{8} +x_{2} x_{9} +2x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -70\right)h_{4}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{6} x_{13} -47\right)h_{3}+\left(x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} -35\right)h_{2}+\left(x_{2} x_{9} +x_{6} x_{13} -24\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x4x11+x6x1335=0x1x8+x2x9+x4x11+x6x1347=0x1x8+x2x9+2x4x11+x6x13+x7x1470=0x1x8+x2x9+x4x11+x5x12+x7x1457=0x1x8+x3x10+x5x12+x7x1444=0x2x9+x6x1324=0x3x10+x5x1230=0x3x1015=0\begin{array}{rcl}x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} -35&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{6} x_{13} -47&=&0\\x_{1} x_{8} +x_{2} x_{9} +2x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -70&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -57&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -44&=&0\\x_{2} x_{9} +x_{6} x_{13} -24&=&0\\x_{3} x_{10} +x_{5} x_{12} -30&=&0\\x_{3} x_{10} -15&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=15h8+30h7+44h6+57h5+70h4+47h3+35h2+24h1e=x1g33+x4g31+x2g24+x6g23+x3g22+x5g21+x7g20f=g20+g21+g22+g23+g24+g31+g33\begin{array}{rcl}h&=&15h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\\e&=&x_{1} g_{33}+x_{4} g_{31}+x_{2} g_{24}+x_{6} g_{23}+x_{3} g_{22}+x_{5} g_{21}+x_{7} g_{20}\\f&=&g_{-20}+g_{-21}+g_{-22}+g_{-23}+g_{-24}+g_{-31}+g_{-33}\end{array}
Matrix form of the system we are trying to solve: (10010101101010110201111011011010101010001000101000010000)[col. vect.]=(3547705744243015)\begin{pmatrix}1 & 0 & 0 & 1 & 0 & 1 & 0\\ 1 & 1 & 0 & 1 & 0 & 1 & 0\\ 1 & 1 & 0 & 2 & 0 & 1 & 1\\ 1 & 1 & 0 & 1 & 1 & 0 & 1\\ 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}35\\ 47\\ 70\\ 57\\ 44\\ 24\\ 30\\ 15\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=15h8+30h7+44h6+57h5+70h4+47h3+35h2+24h1e=x1g33+x4g31+x2g24+x6g23+x3g22+x5g21+x7g20f=x14g20+x12g21+x10g22+x13g23+x9g24+x11g31+x8g33\begin{array}{rcl}h&=&15h_{8}+30h_{7}+44h_{6}+57h_{5}+70h_{4}+47h_{3}+35h_{2}+24h_{1}\\ e&=&x_{1} g_{33}+x_{4} g_{31}+x_{2} g_{24}+x_{6} g_{23}+x_{3} g_{22}+x_{5} g_{21}+x_{7} g_{20}\\ f&=&x_{14} g_{-20}+x_{12} g_{-21}+x_{10} g_{-22}+x_{13} g_{-23}+x_{9} g_{-24}+x_{11} g_{-31}+x_{8} g_{-33}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x4x11+x6x1335=0x1x8+x2x9+x4x11+x6x1347=0x1x8+x2x9+2x4x11+x6x13+x7x1470=0x1x8+x2x9+x4x11+x5x12+x7x1457=0x1x8+x3x10+x5x12+x7x1444=0x2x9+x6x1324=0x3x10+x5x1230=0x3x1015=0\begin{array}{rcl}x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} -35&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{6} x_{13} -47&=&0\\x_{1} x_{8} +x_{2} x_{9} +2x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -70&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -57&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -44&=&0\\x_{2} x_{9} +x_{6} x_{13} -24&=&0\\x_{3} x_{10} +x_{5} x_{12} -30&=&0\\x_{3} x_{10} -15&=&0\\\end{array}

A170A^{70}_1
h-characteristic: (1, 0, 0, 1, 0, 1, 0, 1)
Length of the weight dual to h: 140
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: D5+A3D^{1}_5+A^{1}_3 Containing regular semisimple subalgebra number 2: D7D^{1}_7
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V14ψ+2V13ψ+V12ψ+2V11ψ+2V10ψ+2V9ψ+3V8ψ+4V7ψ+3V6ψ+2V5ψ+3V4ψ+2V3ψ+2V2ψ+2Vψ+V0V_{14\psi}+2V_{13\psi}+V_{12\psi}+2V_{11\psi}+2V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+3V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=14h8+27h7+40h6+52h5+64h4+43h3+32h2+22h1e=10g35+3g34+3g33+8g31+14g29+4g24+18g23+10g20f=g20+g23+g24+g29+g31+g33+g34+g35\begin{array}{rcl}h&=&14h_{8}+27h_{7}+40h_{6}+52h_{5}+64h_{4}+43h_{3}+32h_{2}+22h_{1}\\ e&=&10g_{35}+3g_{34}+3g_{33}+8g_{31}+14g_{29}+4g_{24}+18g_{23}+10g_{20}\\ f&=&g_{-20}+g_{-23}+g_{-24}+g_{-29}+g_{-31}+g_{-33}+g_{-34}+g_{-35}\end{array}
Lie brackets of the above elements.
h=14h8+27h7+40h6+52h5+64h4+43h3+32h2+22h1e=10g35+3g34+3g33+8g31+14g29+4g24+18g23+10g20f=g20+g23+g24+g29+g31+g33+g34+g35\begin{array}{rcl}[e, f]&=&14h_{8}+27h_{7}+40h_{6}+52h_{5}+64h_{4}+43h_{3}+32h_{2}+22h_{1}\\ [h, e]&=&20g_{35}+6g_{34}+6g_{33}+16g_{31}+28g_{29}+8g_{24}+36g_{23}+20g_{20}\\ [h, f]&=&-2g_{-20}-2g_{-23}-2g_{-24}-2g_{-29}-2g_{-31}-2g_{-33}-2g_{-34}-2g_{-35}\end{array}
Centralizer type: 00
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Unknown elements.
h=14h8+27h7+40h6+52h5+64h4+43h3+32h2+22h1e=x4g35+x8g34+x6g33+x1g31+x2g29+x7g24+x3g23+x5g20e=x13g20+x11g23+x15g24+x10g29+x9g31+x14g33+x16g34+x12g35\begin{array}{rcl}h&=&14h_{8}+27h_{7}+40h_{6}+52h_{5}+64h_{4}+43h_{3}+32h_{2}+22h_{1}\\ e&=&x_{4} g_{35}+x_{8} g_{34}+x_{6} g_{33}+x_{1} g_{31}+x_{2} g_{29}+x_{7} g_{24}+x_{3} g_{23}+x_{5} g_{20}\\ f&=&x_{13} g_{-20}+x_{11} g_{-23}+x_{15} g_{-24}+x_{10} g_{-29}+x_{9} g_{-31}+x_{14} g_{-33}+x_{16} g_{-34}+x_{12} g_{-35}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x1014)h8+(x2x10+x4x12+x8x1627)h7+(x2x10+x4x12+x5x13+x6x14+x8x1640)h6+(x1x9+x2x10+x4x12+x5x13+x6x14+x7x15+x8x1652)h5+(2x1x9+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1664)h4+(x1x9+x3x11+x4x12+x6x14+x7x1543)h3+(x1x9+x3x11+x6x14+x8x1632)h2+(x3x11+x7x1522)h1[e,f] - h = \left(x_{2} x_{10} -14\right)h_{8}+\left(x_{2} x_{10} +x_{4} x_{12} +x_{8} x_{16} -27\right)h_{7}+\left(x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{8} x_{16} -40\right)h_{6}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -52\right)h_{5}+\left(2x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -64\right)h_{4}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} -43\right)h_{3}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{8} x_{16} -32\right)h_{2}+\left(x_{3} x_{11} +x_{7} x_{15} -22\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x9+x3x11+x6x14+x8x1632=0x1x9+x3x11+x4x12+x6x14+x7x1543=02x1x9+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1664=0x1x9+x2x10+x4x12+x5x13+x6x14+x7x15+x8x1652=0x2x10+x4x12+x5x13+x6x14+x8x1640=0x2x10+x4x12+x8x1627=0x2x1014=0x3x11+x7x1522=0\begin{array}{rcl}x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{8} x_{16} -32&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} -43&=&0\\2x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -64&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -52&=&0\\x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{8} x_{16} -40&=&0\\x_{2} x_{10} +x_{4} x_{12} +x_{8} x_{16} -27&=&0\\x_{2} x_{10} -14&=&0\\x_{3} x_{11} +x_{7} x_{15} -22&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=14h8+27h7+40h6+52h5+64h4+43h3+32h2+22h1e=x4g35+x8g34+x6g33+x1g31+x2g29+x7g24+x3g23+x5g20f=g20+g23+g24+g29+g31+g33+g34+g35\begin{array}{rcl}h&=&14h_{8}+27h_{7}+40h_{6}+52h_{5}+64h_{4}+43h_{3}+32h_{2}+22h_{1}\\e&=&x_{4} g_{35}+x_{8} g_{34}+x_{6} g_{33}+x_{1} g_{31}+x_{2} g_{29}+x_{7} g_{24}+x_{3} g_{23}+x_{5} g_{20}\\f&=&g_{-20}+g_{-23}+g_{-24}+g_{-29}+g_{-31}+g_{-33}+g_{-34}+g_{-35}\end{array}
Matrix form of the system we are trying to solve: (1010010110110110201111111101111101011101010100010100000000100010)[col. vect.]=(3243645240271422)\begin{pmatrix}1 & 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0\\ 2 & 0 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1\\ 0 & 1 & 0 & 1 & 1 & 1 & 0 & 1\\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}32\\ 43\\ 64\\ 52\\ 40\\ 27\\ 14\\ 22\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=14h8+27h7+40h6+52h5+64h4+43h3+32h2+22h1e=x4g35+x8g34+x6g33+x1g31+x2g29+x7g24+x3g23+x5g20f=x13g20+x11g23+x15g24+x10g29+x9g31+x14g33+x16g34+x12g35\begin{array}{rcl}h&=&14h_{8}+27h_{7}+40h_{6}+52h_{5}+64h_{4}+43h_{3}+32h_{2}+22h_{1}\\ e&=&x_{4} g_{35}+x_{8} g_{34}+x_{6} g_{33}+x_{1} g_{31}+x_{2} g_{29}+x_{7} g_{24}+x_{3} g_{23}+x_{5} g_{20}\\ f&=&x_{13} g_{-20}+x_{11} g_{-23}+x_{15} g_{-24}+x_{10} g_{-29}+x_{9} g_{-31}+x_{14} g_{-33}+x_{16} g_{-34}+x_{12} g_{-35}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x9+x3x11+x6x14+x8x1632=0x1x9+x3x11+x4x12+x6x14+x7x1543=02x1x9+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1664=0x1x9+x2x10+x4x12+x5x13+x6x14+x7x15+x8x1652=0x2x10+x4x12+x5x13+x6x14+x8x1640=0x2x10+x4x12+x8x1627=0x2x1014=0x3x11+x7x1522=0\begin{array}{rcl}x_{1} x_{9} +x_{3} x_{11} +x_{6} x_{14} +x_{8} x_{16} -32&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} -43&=&0\\2x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -64&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -52&=&0\\x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{8} x_{16} -40&=&0\\x_{2} x_{10} +x_{4} x_{12} +x_{8} x_{16} -27&=&0\\x_{2} x_{10} -14&=&0\\x_{3} x_{11} +x_{7} x_{15} -22&=&0\\\end{array}

A164A^{64}_1
h-characteristic: (0, 0, 0, 0, 2, 0, 0, 2)
Length of the weight dual to h: 128
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: D5+A2D^{1}_5+A^{1}_2 Containing regular semisimple subalgebra number 2: E7+A1E^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 3: D6+2A1D^{1}_6+2A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V14ψ+2V12ψ+7V10ψ+5V8ψ+5V6ψ+5V4ψ+8V2ψ+V0V_{14\psi}+2V_{12\psi}+7V_{10\psi}+5V_{8\psi}+5V_{6\psi}+5V_{4\psi}+8V_{2\psi}+V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=14h8+26h7+38h6+50h5+60h4+40h3+30h2+20h1e=8g58+2g37+18g34+2g33+10g32+10g19+14g8f=g8+g19+g32+g33+g34+g37+g58\begin{array}{rcl}h&=&14h_{8}+26h_{7}+38h_{6}+50h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\\ e&=&8g_{58}+2g_{37}+18g_{34}+2g_{33}+10g_{32}+10g_{19}+14g_{8}\\ f&=&g_{-8}+g_{-19}+g_{-32}+g_{-33}+g_{-34}+g_{-37}+g_{-58}\end{array}
Lie brackets of the above elements.
h=14h8+26h7+38h6+50h5+60h4+40h3+30h2+20h1e=8g58+2g37+18g34+2g33+10g32+10g19+14g8f=g8+g19+g32+g33+g34+g37+g58\begin{array}{rcl}[e, f]&=&14h_{8}+26h_{7}+38h_{6}+50h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\\ [h, e]&=&16g_{58}+4g_{37}+36g_{34}+4g_{33}+20g_{32}+20g_{19}+28g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-19}-2g_{-32}-2g_{-33}-2g_{-34}-2g_{-37}-2g_{-58}\end{array}
Centralizer type: 00
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Unknown elements.
h=14h8+26h7+38h6+50h5+60h4+40h3+30h2+20h1e=x1g58+x6g37+x3g34+x7g33+x4g32+x5g19+x2g8e=x9g8+x12g19+x11g32+x14g33+x10g34+x13g37+x8g58\begin{array}{rcl}h&=&14h_{8}+26h_{7}+38h_{6}+50h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\\ e&=&x_{1} g_{58}+x_{6} g_{37}+x_{3} g_{34}+x_{7} g_{33}+x_{4} g_{32}+x_{5} g_{19}+x_{2} g_{8}\\ f&=&x_{9} g_{-8}+x_{12} g_{-19}+x_{11} g_{-32}+x_{14} g_{-33}+x_{10} g_{-34}+x_{13} g_{-37}+x_{8} g_{-58}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x914)h8+(x1x8+x3x1026)h7+(x1x8+x3x10+x4x11+x7x1438)h6+(x1x8+x3x10+x4x11+x5x12+x6x13+x7x1450)h5+(2x1x8+x3x10+x4x11+x5x12+2x6x13+x7x1460)h4+(2x1x8+x4x11+x5x12+x6x13+x7x1440)h3+(x1x8+x3x10+x6x13+x7x1430)h2+(x1x8+x4x11+x6x1320)h1[e,f] - h = \left(x_{2} x_{9} -14\right)h_{8}+\left(x_{1} x_{8} +x_{3} x_{10} -26\right)h_{7}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{7} x_{14} -38\right)h_{6}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -50\right)h_{5}+\left(2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -60\right)h_{4}+\left(2x_{1} x_{8} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -40\right)h_{3}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -30\right)h_{2}+\left(x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} -20\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x4x11+x6x1320=0x1x8+x3x10+x6x13+x7x1430=02x1x8+x4x11+x5x12+x6x13+x7x1440=02x1x8+x3x10+x4x11+x5x12+2x6x13+x7x1460=0x1x8+x3x10+x4x11+x5x12+x6x13+x7x1450=0x1x8+x3x10+x4x11+x7x1438=0x1x8+x3x1026=0x2x914=0\begin{array}{rcl}x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} -20&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -30&=&0\\2x_{1} x_{8} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -40&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -60&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -50&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{7} x_{14} -38&=&0\\x_{1} x_{8} +x_{3} x_{10} -26&=&0\\x_{2} x_{9} -14&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=14h8+26h7+38h6+50h5+60h4+40h3+30h2+20h1e=x1g58+x6g37+x3g34+x7g33+x4g32+x5g19+x2g8f=g8+g19+g32+g33+g34+g37+g58\begin{array}{rcl}h&=&14h_{8}+26h_{7}+38h_{6}+50h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\\e&=&x_{1} g_{58}+x_{6} g_{37}+x_{3} g_{34}+x_{7} g_{33}+x_{4} g_{32}+x_{5} g_{19}+x_{2} g_{8}\\f&=&g_{-8}+g_{-19}+g_{-32}+g_{-33}+g_{-34}+g_{-37}+g_{-58}\end{array}
Matrix form of the system we are trying to solve: (10010101010011200111120111211011111101100110100000100000)[col. vect.]=(2030406050382614)\begin{pmatrix}1 & 0 & 0 & 1 & 0 & 1 & 0\\ 1 & 0 & 1 & 0 & 0 & 1 & 1\\ 2 & 0 & 0 & 1 & 1 & 1 & 1\\ 2 & 0 & 1 & 1 & 1 & 2 & 1\\ 1 & 0 & 1 & 1 & 1 & 1 & 1\\ 1 & 0 & 1 & 1 & 0 & 0 & 1\\ 1 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}20\\ 30\\ 40\\ 60\\ 50\\ 38\\ 26\\ 14\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=14h8+26h7+38h6+50h5+60h4+40h3+30h2+20h1e=x1g58+x6g37+x3g34+x7g33+x4g32+x5g19+x2g8f=x9g8+x12g19+x11g32+x14g33+x10g34+x13g37+x8g58\begin{array}{rcl}h&=&14h_{8}+26h_{7}+38h_{6}+50h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\\ e&=&x_{1} g_{58}+x_{6} g_{37}+x_{3} g_{34}+x_{7} g_{33}+x_{4} g_{32}+x_{5} g_{19}+x_{2} g_{8}\\ f&=&x_{9} g_{-8}+x_{12} g_{-19}+x_{11} g_{-32}+x_{14} g_{-33}+x_{10} g_{-34}+x_{13} g_{-37}+x_{8} g_{-58}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x4x11+x6x1320=0x1x8+x3x10+x6x13+x7x1430=02x1x8+x4x11+x5x12+x6x13+x7x1440=02x1x8+x3x10+x4x11+x5x12+2x6x13+x7x1460=0x1x8+x3x10+x4x11+x5x12+x6x13+x7x1450=0x1x8+x3x10+x4x11+x7x1438=0x1x8+x3x1026=0x2x914=0\begin{array}{rcl}x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} -20&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -30&=&0\\2x_{1} x_{8} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -40&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -60&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -50&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{7} x_{14} -38&=&0\\x_{1} x_{8} +x_{3} x_{10} -26&=&0\\x_{2} x_{9} -14&=&0\\\end{array}

A163A^{63}_1
h-characteristic: (0, 0, 0, 1, 0, 1, 0, 2)
Length of the weight dual to h: 126
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: E7E^{1}_7 Containing regular semisimple subalgebra number 2: D6+A1D^{1}_6+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V14ψ+V12ψ+2V11ψ+4V10ψ+4V9ψ+2V8ψ+2V7ψ+3V6ψ+2V5ψ+2V4ψ+4V3ψ+4V2ψ+2Vψ+3V0V_{14\psi}+V_{12\psi}+2V_{11\psi}+4V_{10\psi}+4V_{9\psi}+2V_{8\psi}+2V_{7\psi}+3V_{6\psi}+2V_{5\psi}+2V_{4\psi}+4V_{3\psi}+4V_{2\psi}+2V_{\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=14h8+26h7+38h6+49h5+60h4+40h3+30h2+20h1e=272g44+132g416g40+15g3852g37+32g35+52g34392g339g32+232g31+232g28+92g2752g26+272g20+14g15f=g8+g15+g20+g26+g28+g37+g38+g40+g41+g44+g47\begin{array}{rcl}h&=&14h_{8}+26h_{7}+38h_{6}+49h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\\ e&=&27/2g_{44}+13/2g_{41}-6g_{40}+15g_{38}-5/2g_{37}+3/2g_{35}+5/2g_{34}-39/2g_{33}-9g_{32}+23/2g_{31}+23/2g_{28}+9/2g_{27}-5/2g_{26}+27/2g_{20}+14g_{15}\\ f&=&g_{-8}+g_{-15}+g_{-20}+g_{-26}+g_{-28}+g_{-37}+g_{-38}+g_{-40}+g_{-41}+g_{-44}+g_{-47}\end{array}
Lie brackets of the above elements.
h=14h8+26h7+38h6+49h5+60h4+40h3+30h2+20h1e=272g44+132g416g40+15g3852g37+32g35+52g34392g339g32+232g31+232g28+92g2752g26+272g20+14g15f=g8+g15+g20+g26+g28+g37+g38+g40+g41+g44+g47\begin{array}{rcl}[e, f]&=&14h_{8}+26h_{7}+38h_{6}+49h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\\ [h, e]&=&27g_{44}+13g_{41}-12g_{40}+30g_{38}-5g_{37}+3g_{35}+5g_{34}-39g_{33}-18g_{32}+23g_{31}+23g_{28}+9g_{27}-5g_{26}+27g_{20}+28g_{15}\\ [h, f]&=&-2g_{-8}-2g_{-15}-2g_{-20}-2g_{-26}-2g_{-28}-2g_{-37}-2g_{-38}-2g_{-40}-2g_{-41}-2g_{-44}-2g_{-47}\end{array}
Centralizer type: A1A_1
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Unknown elements.
h=14h8+26h7+38h6+49h5+60h4+40h3+30h2+20h1e=x2g47+x3g44+x7g41+x5g40+x6g38+x8g37+x10g35+x12g34+x11g33+x9g32+x13g31+x15g28+x14g27+x16g26+x17g20+x1g15+x4g8e=x21g8+x18g15+x34g20+x33g26+x31g27+x32g28+x30g31+x26g32+x28g33+x29g34+x27g35+x25g37+x23g38+x22g40+x24g41+x20g44+x19g47\begin{array}{rcl}h&=&14h_{8}+26h_{7}+38h_{6}+49h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\\ e&=&x_{2} g_{47}+x_{3} g_{44}+x_{7} g_{41}+x_{5} g_{40}+x_{6} g_{38}+x_{8} g_{37}+x_{10} g_{35}+x_{12} g_{34}+x_{11} g_{33}+x_{9} g_{32}+x_{13} g_{31}+x_{15} g_{28}+x_{14} g_{27}+x_{16} g_{26}+x_{17} g_{20}+x_{1} g_{15}+x_{4} g_{8}\\ f&=&x_{21} g_{-8}+x_{18} g_{-15}+x_{34} g_{-20}+x_{33} g_{-26}+x_{31} g_{-27}+x_{32} g_{-28}+x_{30} g_{-31}+x_{26} g_{-32}+x_{28} g_{-33}+x_{29} g_{-34}+x_{27} g_{-35}+x_{25} g_{-37}+x_{23} g_{-38}+x_{22} g_{-40}+x_{24} g_{-41}+x_{20} g_{-44}+x_{19} g_{-47}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x29x3x30+x5x32+x6x33+x9x34)g9+(x1x21x2x23x5x26x7x28x10x31x12x33x15x34)g7+(x3x25+x7x29+x10x32+x11x33+x14x34)g3+(x2x22+x6x26+x7x27+x11x31+x12x32+x16x34)g2+(x2x24+x5x27+x6x28+x8x30+x9x31)g1+(x1x18+x4x2114)h8+(x1x18+x2x19+x5x22+x7x24+x10x27+x12x29+x15x3226)h7+(x2x19+x5x22+x6x23+x7x24+x9x26+x10x27+x11x28+x12x29+x14x31+x15x32+x16x33+x17x3438)h6+(x2x19+x3x20+x5x22+x6x23+x7x24+x8x25+x9x26+x10x27+x11x28+x12x29+x13x30+x14x31+x15x32+x16x33+x17x3449)h5+(x2x19+2x3x20+x5x22+x6x23+x7x24+2x8x25+x9x26+x10x27+x11x28+x12x29+2x13x30+x14x31+x15x32+x16x33+x17x3460)h4+(x2x19+2x3x20+x5x22+x6x23+x7x24+x8x25+x9x26+x10x27+x11x28+x13x30+x14x3140)h3+(x2x19+x3x20+x6x23+x7x24+x8x25+x11x28+x12x29+x13x30+x16x3330)h2+(x2x19+x3x20+x5x22+x6x23+x8x25+x9x2620)h1+(x7x19+x10x22+x11x23+x13x25+x14x26)g1+(x5x19+x9x23+x10x24+x14x28+x15x29+x17x33)g2+(x8x20+x12x24+x15x27+x16x28+x17x31)g3+(x4x18x6x19x9x22x11x24x14x27x16x29x17x32)g7+(x12x19x13x20+x15x22+x16x23+x17x26)g9[e,f] - h = \left(x_{2} x_{29} -x_{3} x_{30} +x_{5} x_{32} +x_{6} x_{33} +x_{9} x_{34} \right)g_{9}+\left(x_{1} x_{21} -x_{2} x_{23} -x_{5} x_{26} -x_{7} x_{28} -x_{10} x_{31} -x_{12} x_{33} -x_{15} x_{34} \right)g_{7}+\left(x_{3} x_{25} +x_{7} x_{29} +x_{10} x_{32} +x_{11} x_{33} +x_{14} x_{34} \right)g_{3}+\left(x_{2} x_{22} +x_{6} x_{26} +x_{7} x_{27} +x_{11} x_{31} +x_{12} x_{32} +x_{16} x_{34} \right)g_{2}+\left(x_{2} x_{24} +x_{5} x_{27} +x_{6} x_{28} +x_{8} x_{30} +x_{9} x_{31} \right)g_{1}+\left(x_{1} x_{18} +x_{4} x_{21} -14\right)h_{8}+\left(x_{1} x_{18} +x_{2} x_{19} +x_{5} x_{22} +x_{7} x_{24} +x_{10} x_{27} +x_{12} x_{29} +x_{15} x_{32} -26\right)h_{7}+\left(x_{2} x_{19} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{12} x_{29} +x_{14} x_{31} +x_{15} x_{32} +x_{16} x_{33} +x_{17} x_{34} -38\right)h_{6}+\left(x_{2} x_{19} +x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +x_{8} x_{25} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{12} x_{29} +x_{13} x_{30} +x_{14} x_{31} +x_{15} x_{32} +x_{16} x_{33} +x_{17} x_{34} -49\right)h_{5}+\left(x_{2} x_{19} +2x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +2x_{8} x_{25} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{12} x_{29} +2x_{13} x_{30} +x_{14} x_{31} +x_{15} x_{32} +x_{16} x_{33} +x_{17} x_{34} -60\right)h_{4}+\left(x_{2} x_{19} +2x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +x_{8} x_{25} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{13} x_{30} +x_{14} x_{31} -40\right)h_{3}+\left(x_{2} x_{19} +x_{3} x_{20} +x_{6} x_{23} +x_{7} x_{24} +x_{8} x_{25} +x_{11} x_{28} +x_{12} x_{29} +x_{13} x_{30} +x_{16} x_{33} -30\right)h_{2}+\left(x_{2} x_{19} +x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{8} x_{25} +x_{9} x_{26} -20\right)h_{1}+\left(x_{7} x_{19} +x_{10} x_{22} +x_{11} x_{23} +x_{13} x_{25} +x_{14} x_{26} \right)g_{-1}+\left(x_{5} x_{19} +x_{9} x_{23} +x_{10} x_{24} +x_{14} x_{28} +x_{15} x_{29} +x_{17} x_{33} \right)g_{-2}+\left(x_{8} x_{20} +x_{12} x_{24} +x_{15} x_{27} +x_{16} x_{28} +x_{17} x_{31} \right)g_{-3}+\left(x_{4} x_{18} -x_{6} x_{19} -x_{9} x_{22} -x_{11} x_{24} -x_{14} x_{27} -x_{16} x_{29} -x_{17} x_{32} \right)g_{-7}+\left(x_{12} x_{19} -x_{13} x_{20} +x_{15} x_{22} +x_{16} x_{23} +x_{17} x_{26} \right)g_{-9}
The polynomial system that corresponds to finding the h, e, f triple:
x1x18+x2x19+x5x22+x7x24+x10x27+x12x29+x15x3226=0x1x18+x4x2114=0x1x21x2x23x5x26x7x28x10x31x12x33x15x34=0x2x19+x3x20+x5x22+x6x23+x8x25+x9x2620=0x2x19+x3x20+x6x23+x7x24+x8x25+x11x28+x12x29+x13x30+x16x3330=0x2x19+2x3x20+x5x22+x6x23+x7x24+x8x25+x9x26+x10x27+x11x28+x13x30+x14x3140=0x2x19+2x3x20+x5x22+x6x23+x7x24+2x8x25+x9x26+x10x27+x11x28+x12x29+2x13x30+x14x31+x15x32+x16x33+x17x3460=0x2x19+x3x20+x5x22+x6x23+x7x24+x8x25+x9x26+x10x27+x11x28+x12x29+x13x30+x14x31+x15x32+x16x33+x17x3449=0x2x19+x5x22+x6x23+x7x24+x9x26+x10x27+x11x28+x12x29+x14x31+x15x32+x16x33+x17x3438=0x2x22+x6x26+x7x27+x11x31+x12x32+x16x34=0x2x24+x5x27+x6x28+x8x30+x9x31=0x2x29x3x30+x5x32+x6x33+x9x34=0x3x25+x7x29+x10x32+x11x33+x14x34=0x4x18x6x19x9x22x11x24x14x27x16x29x17x32=0x5x19+x9x23+x10x24+x14x28+x15x29+x17x33=0x7x19+x10x22+x11x23+x13x25+x14x26=0x8x20+x12x24+x15x27+x16x28+x17x31=0x12x19x13x20+x15x22+x16x23+x17x26=0\begin{array}{rcl}x_{1} x_{18} +x_{2} x_{19} +x_{5} x_{22} +x_{7} x_{24} +x_{10} x_{27} +x_{12} x_{29} +x_{15} x_{32} -26&=&0\\x_{1} x_{18} +x_{4} x_{21} -14&=&0\\x_{1} x_{21} -x_{2} x_{23} -x_{5} x_{26} -x_{7} x_{28} -x_{10} x_{31} -x_{12} x_{33} -x_{15} x_{34} &=&0\\x_{2} x_{19} +x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{8} x_{25} +x_{9} x_{26} -20&=&0\\x_{2} x_{19} +x_{3} x_{20} +x_{6} x_{23} +x_{7} x_{24} +x_{8} x_{25} +x_{11} x_{28} +x_{12} x_{29} +x_{13} x_{30} +x_{16} x_{33} -30&=&0\\x_{2} x_{19} +2x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +x_{8} x_{25} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{13} x_{30} +x_{14} x_{31} -40&=&0\\x_{2} x_{19} +2x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +2x_{8} x_{25} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{12} x_{29} +2x_{13} x_{30} +x_{14} x_{31} +x_{15} x_{32} +x_{16} x_{33} +x_{17} x_{34} -60&=&0\\x_{2} x_{19} +x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +x_{8} x_{25} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{12} x_{29} +x_{13} x_{30} +x_{14} x_{31} +x_{15} x_{32} +x_{16} x_{33} +x_{17} x_{34} -49&=&0\\x_{2} x_{19} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{12} x_{29} +x_{14} x_{31} +x_{15} x_{32} +x_{16} x_{33} +x_{17} x_{34} -38&=&0\\x_{2} x_{22} +x_{6} x_{26} +x_{7} x_{27} +x_{11} x_{31} +x_{12} x_{32} +x_{16} x_{34} &=&0\\x_{2} x_{24} +x_{5} x_{27} +x_{6} x_{28} +x_{8} x_{30} +x_{9} x_{31} &=&0\\x_{2} x_{29} -x_{3} x_{30} +x_{5} x_{32} +x_{6} x_{33} +x_{9} x_{34} &=&0\\x_{3} x_{25} +x_{7} x_{29} +x_{10} x_{32} +x_{11} x_{33} +x_{14} x_{34} &=&0\\x_{4} x_{18} -x_{6} x_{19} -x_{9} x_{22} -x_{11} x_{24} -x_{14} x_{27} -x_{16} x_{29} -x_{17} x_{32} &=&0\\x_{5} x_{19} +x_{9} x_{23} +x_{10} x_{24} +x_{14} x_{28} +x_{15} x_{29} +x_{17} x_{33} &=&0\\x_{7} x_{19} +x_{10} x_{22} +x_{11} x_{23} +x_{13} x_{25} +x_{14} x_{26} &=&0\\x_{8} x_{20} +x_{12} x_{24} +x_{15} x_{27} +x_{16} x_{28} +x_{17} x_{31} &=&0\\x_{12} x_{19} -x_{13} x_{20} +x_{15} x_{22} +x_{16} x_{23} +x_{17} x_{26} &=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=14h8+26h7+38h6+49h5+60h4+40h3+30h2+20h1e=x2g47+x3g44+x7g41+x5g40+x6g38+x8g37+x10g35+x12g34+x11g33+x9g32+x13g31+x15g28+x14g27+x16g26+x17g20+x1g15+x4g8f=g8+g15+g20+g26+g28+g37+g38+g40+g41+g44+g47\begin{array}{rcl}h&=&14h_{8}+26h_{7}+38h_{6}+49h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\\e&=&x_{2} g_{47}+x_{3} g_{44}+x_{7} g_{41}+x_{5} g_{40}+x_{6} g_{38}+x_{8} g_{37}+x_{10} g_{35}+x_{12} g_{34}+x_{11} g_{33}+x_{9} g_{32}+x_{13} g_{31}+x_{15} g_{28}+x_{14} g_{27}+x_{16} g_{26}+x_{17} g_{20}+x_{1} g_{15}+x_{4} g_{8}\\f&=&g_{-8}+g_{-15}+g_{-20}+g_{-26}+g_{-28}+g_{-37}+g_{-38}+g_{-40}+g_{-41}+g_{-44}+g_{-47}\end{array}
Matrix form of the system we are trying to solve: (110010100000001001001000000000000011000000000100100011011010000000000110011100000001001201111000000000012011120000001110110111100000011101001110000000111010000000001000100100000000000000000100000011001000000101001010000010000100011000000100001100100000000000000100110100000000000100010000000000000000110110)[col. vect.]=(26140203040604938000000000)\begin{pmatrix}1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0\\ 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 2 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 2 & 0 & 1 & 1 & 1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1\\ 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1\\ 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 1 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}26\\ 14\\ 0\\ 20\\ 30\\ 40\\ 60\\ 49\\ 38\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=14h8+26h7+38h6+49h5+60h4+40h3+30h2+20h1e=x2g47+x3g44+x7g41+x5g40+x6g38+x8g37+x10g35+x12g34+x11g33+x9g32+x13g31+x15g28+x14g27+x16g26+x17g20+x1g15+x4g8f=x21g8+x18g15+x34g20+x33g26+x31g27+x32g28+x30g31+x26g32+x28g33+x29g34+x27g35+x25g37+x23g38+x22g40+x24g41+x20g44+x19g47\begin{array}{rcl}h&=&14h_{8}+26h_{7}+38h_{6}+49h_{5}+60h_{4}+40h_{3}+30h_{2}+20h_{1}\\ e&=&x_{2} g_{47}+x_{3} g_{44}+x_{7} g_{41}+x_{5} g_{40}+x_{6} g_{38}+x_{8} g_{37}+x_{10} g_{35}+x_{12} g_{34}+x_{11} g_{33}+x_{9} g_{32}+x_{13} g_{31}+x_{15} g_{28}+x_{14} g_{27}+x_{16} g_{26}+x_{17} g_{20}+x_{1} g_{15}+x_{4} g_{8}\\ f&=&x_{21} g_{-8}+x_{18} g_{-15}+x_{34} g_{-20}+x_{33} g_{-26}+x_{31} g_{-27}+x_{32} g_{-28}+x_{30} g_{-31}+x_{26} g_{-32}+x_{28} g_{-33}+x_{29} g_{-34}+x_{27} g_{-35}+x_{25} g_{-37}+x_{23} g_{-38}+x_{22} g_{-40}+x_{24} g_{-41}+x_{20} g_{-44}+x_{19} g_{-47}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x18+x2x19+x5x22+x7x24+x10x27+x12x29+x15x3226=0x1x18+x4x2114=0x1x21x2x23x5x26x7x28x10x31x12x33x15x34=0x2x19+x3x20+x5x22+x6x23+x8x25+x9x2620=0x2x19+x3x20+x6x23+x7x24+x8x25+x11x28+x12x29+x13x30+x16x3330=0x2x19+2x3x20+x5x22+x6x23+x7x24+x8x25+x9x26+x10x27+x11x28+x13x30+x14x3140=0x2x19+2x3x20+x5x22+x6x23+x7x24+2x8x25+x9x26+x10x27+x11x28+x12x29+2x13x30+x14x31+x15x32+x16x33+x17x3460=0x2x19+x3x20+x5x22+x6x23+x7x24+x8x25+x9x26+x10x27+x11x28+x12x29+x13x30+x14x31+x15x32+x16x33+x17x3449=0x2x19+x5x22+x6x23+x7x24+x9x26+x10x27+x11x28+x12x29+x14x31+x15x32+x16x33+x17x3438=0x2x22+x6x26+x7x27+x11x31+x12x32+x16x34=0x2x24+x5x27+x6x28+x8x30+x9x31=0x2x29x3x30+x5x32+x6x33+x9x34=0x3x25+x7x29+x10x32+x11x33+x14x34=0x4x18x6x19x9x22x11x24x14x27x16x29x17x32=0x5x19+x9x23+x10x24+x14x28+x15x29+x17x33=0x7x19+x10x22+x11x23+x13x25+x14x26=0x8x20+x12x24+x15x27+x16x28+x17x31=0x12x19x13x20+x15x22+x16x23+x17x26=0\begin{array}{rcl}x_{1} x_{18} +x_{2} x_{19} +x_{5} x_{22} +x_{7} x_{24} +x_{10} x_{27} +x_{12} x_{29} +x_{15} x_{32} -26&=&0\\x_{1} x_{18} +x_{4} x_{21} -14&=&0\\x_{1} x_{21} -x_{2} x_{23} -x_{5} x_{26} -x_{7} x_{28} -x_{10} x_{31} -x_{12} x_{33} -x_{15} x_{34} &=&0\\x_{2} x_{19} +x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{8} x_{25} +x_{9} x_{26} -20&=&0\\x_{2} x_{19} +x_{3} x_{20} +x_{6} x_{23} +x_{7} x_{24} +x_{8} x_{25} +x_{11} x_{28} +x_{12} x_{29} +x_{13} x_{30} +x_{16} x_{33} -30&=&0\\x_{2} x_{19} +2x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +x_{8} x_{25} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{13} x_{30} +x_{14} x_{31} -40&=&0\\x_{2} x_{19} +2x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +2x_{8} x_{25} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{12} x_{29} +2x_{13} x_{30} +x_{14} x_{31} +x_{15} x_{32} +x_{16} x_{33} +x_{17} x_{34} -60&=&0\\x_{2} x_{19} +x_{3} x_{20} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +x_{8} x_{25} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{12} x_{29} +x_{13} x_{30} +x_{14} x_{31} +x_{15} x_{32} +x_{16} x_{33} +x_{17} x_{34} -49&=&0\\x_{2} x_{19} +x_{5} x_{22} +x_{6} x_{23} +x_{7} x_{24} +x_{9} x_{26} +x_{10} x_{27} +x_{11} x_{28} +x_{12} x_{29} +x_{14} x_{31} +x_{15} x_{32} +x_{16} x_{33} +x_{17} x_{34} -38&=&0\\x_{2} x_{22} +x_{6} x_{26} +x_{7} x_{27} +x_{11} x_{31} +x_{12} x_{32} +x_{16} x_{34} &=&0\\x_{2} x_{24} +x_{5} x_{27} +x_{6} x_{28} +x_{8} x_{30} +x_{9} x_{31} &=&0\\x_{2} x_{29} -x_{3} x_{30} +x_{5} x_{32} +x_{6} x_{33} +x_{9} x_{34} &=&0\\x_{3} x_{25} +x_{7} x_{29} +x_{10} x_{32} +x_{11} x_{33} +x_{14} x_{34} &=&0\\x_{4} x_{18} -x_{6} x_{19} -x_{9} x_{22} -x_{11} x_{24} -x_{14} x_{27} -x_{16} x_{29} -x_{17} x_{32} &=&0\\x_{5} x_{19} +x_{9} x_{23} +x_{10} x_{24} +x_{14} x_{28} +x_{15} x_{29} +x_{17} x_{33} &=&0\\x_{7} x_{19} +x_{10} x_{22} +x_{11} x_{23} +x_{13} x_{25} +x_{14} x_{26} &=&0\\x_{8} x_{20} +x_{12} x_{24} +x_{15} x_{27} +x_{16} x_{28} +x_{17} x_{31} &=&0\\x_{12} x_{19} -x_{13} x_{20} +x_{15} x_{22} +x_{16} x_{23} +x_{17} x_{26} &=&0\\\end{array}

A162A^{62}_1
h-characteristic: (0, 1, 1, 0, 0, 0, 1, 2)
Length of the weight dual to h: 124
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: D5+2A1D^{1}_5+2A^{1}_1 Containing regular semisimple subalgebra number 2: D6D^{1}_6
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V14ψ+4V11ψ+2V10ψ+4V9ψ+5V8ψ+2V6ψ+4V5ψ+4V3ψ+6V2ψ+6V0V_{14\psi}+4V_{11\psi}+2V_{10\psi}+4V_{9\psi}+5V_{8\psi}+2V_{6\psi}+4V_{5\psi}+4V_{3\psi}+6V_{2\psi}+6V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=14h8+26h7+37h6+48h5+59h4+40h3+30h2+20h1e=10g46+g45+18g40+8g34+g30+10g17+14g8f=g8+g17+g30+g34+g40+g45+g46\begin{array}{rcl}h&=&14h_{8}+26h_{7}+37h_{6}+48h_{5}+59h_{4}+40h_{3}+30h_{2}+20h_{1}\\ e&=&10g_{46}+g_{45}+18g_{40}+8g_{34}+g_{30}+10g_{17}+14g_{8}\\ f&=&g_{-8}+g_{-17}+g_{-30}+g_{-34}+g_{-40}+g_{-45}+g_{-46}\end{array}
Lie brackets of the above elements.
h=14h8+26h7+37h6+48h5+59h4+40h3+30h2+20h1e=10g46+g45+18g40+8g34+g30+10g17+14g8f=g8+g17+g30+g34+g40+g45+g46\begin{array}{rcl}[e, f]&=&14h_{8}+26h_{7}+37h_{6}+48h_{5}+59h_{4}+40h_{3}+30h_{2}+20h_{1}\\ [h, e]&=&20g_{46}+2g_{45}+36g_{40}+16g_{34}+2g_{30}+20g_{17}+28g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-17}-2g_{-30}-2g_{-34}-2g_{-40}-2g_{-45}-2g_{-46}\end{array}
Centralizer type: 2A12A_1
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Unknown elements.
h=14h8+26h7+37h6+48h5+59h4+40h3+30h2+20h1e=x4g46+x6g45+x3g40+x1g34+x7g30+x5g17+x2g8e=x9g8+x12g17+x14g30+x8g34+x10g40+x13g45+x11g46\begin{array}{rcl}h&=&14h_{8}+26h_{7}+37h_{6}+48h_{5}+59h_{4}+40h_{3}+30h_{2}+20h_{1}\\ e&=&x_{4} g_{46}+x_{6} g_{45}+x_{3} g_{40}+x_{1} g_{34}+x_{7} g_{30}+x_{5} g_{17}+x_{2} g_{8}\\ f&=&x_{9} g_{-8}+x_{12} g_{-17}+x_{14} g_{-30}+x_{8} g_{-34}+x_{10} g_{-40}+x_{13} g_{-45}+x_{11} g_{-46}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x914)h8+(x1x8+x3x1026)h7+(x1x8+x3x10+x4x11+x6x1337)h6+(x1x8+x3x10+2x4x11+x6x13+x7x1448)h5+(x1x8+x3x10+2x4x11+x5x12+2x6x13+x7x1459)h4+(x3x10+x4x11+x5x12+x6x13+x7x1440)h3+(x1x8+x4x11+x5x12+x6x13+x7x1430)h2+(x3x10+x6x13+x7x1420)h1[e,f] - h = \left(x_{2} x_{9} -14\right)h_{8}+\left(x_{1} x_{8} +x_{3} x_{10} -26\right)h_{7}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -37\right)h_{6}+\left(x_{1} x_{8} +x_{3} x_{10} +2x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -48\right)h_{5}+\left(x_{1} x_{8} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -59\right)h_{4}+\left(x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -40\right)h_{3}+\left(x_{1} x_{8} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -30\right)h_{2}+\left(x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -20\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x4x11+x5x12+x6x13+x7x1430=0x1x8+x3x10+2x4x11+x5x12+2x6x13+x7x1459=0x1x8+x3x10+2x4x11+x6x13+x7x1448=0x1x8+x3x10+x4x11+x6x1337=0x1x8+x3x1026=0x2x914=0x3x10+x6x13+x7x1420=0x3x10+x4x11+x5x12+x6x13+x7x1440=0\begin{array}{rcl}x_{1} x_{8} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -30&=&0\\x_{1} x_{8} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -59&=&0\\x_{1} x_{8} +x_{3} x_{10} +2x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -48&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -37&=&0\\x_{1} x_{8} +x_{3} x_{10} -26&=&0\\x_{2} x_{9} -14&=&0\\x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -20&=&0\\x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -40&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=14h8+26h7+37h6+48h5+59h4+40h3+30h2+20h1e=x4g46+x6g45+x3g40+x1g34+x7g30+x5g17+x2g8f=g8+g17+g30+g34+g40+g45+g46\begin{array}{rcl}h&=&14h_{8}+26h_{7}+37h_{6}+48h_{5}+59h_{4}+40h_{3}+30h_{2}+20h_{1}\\e&=&x_{4} g_{46}+x_{6} g_{45}+x_{3} g_{40}+x_{1} g_{34}+x_{7} g_{30}+x_{5} g_{17}+x_{2} g_{8}\\f&=&g_{-8}+g_{-17}+g_{-30}+g_{-34}+g_{-40}+g_{-45}+g_{-46}\end{array}
Matrix form of the system we are trying to solve: (10011111012121101201110110101010000010000000100110011111)[col. vect.]=(3059483726142040)\begin{pmatrix}1 & 0 & 0 & 1 & 1 & 1 & 1\\ 1 & 0 & 1 & 2 & 1 & 2 & 1\\ 1 & 0 & 1 & 2 & 0 & 1 & 1\\ 1 & 0 & 1 & 1 & 0 & 1 & 0\\ 1 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 1 & 1\\ 0 & 0 & 1 & 1 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}30\\ 59\\ 48\\ 37\\ 26\\ 14\\ 20\\ 40\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=14h8+26h7+37h6+48h5+59h4+40h3+30h2+20h1e=x4g46+x6g45+x3g40+x1g34+x7g30+x5g17+x2g8f=x9g8+x12g17+x14g30+x8g34+x10g40+x13g45+x11g46\begin{array}{rcl}h&=&14h_{8}+26h_{7}+37h_{6}+48h_{5}+59h_{4}+40h_{3}+30h_{2}+20h_{1}\\ e&=&x_{4} g_{46}+x_{6} g_{45}+x_{3} g_{40}+x_{1} g_{34}+x_{7} g_{30}+x_{5} g_{17}+x_{2} g_{8}\\ f&=&x_{9} g_{-8}+x_{12} g_{-17}+x_{14} g_{-30}+x_{8} g_{-34}+x_{10} g_{-40}+x_{13} g_{-45}+x_{11} g_{-46}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x4x11+x5x12+x6x13+x7x1430=0x1x8+x3x10+2x4x11+x5x12+2x6x13+x7x1459=0x1x8+x3x10+2x4x11+x6x13+x7x1448=0x1x8+x3x10+x4x11+x6x1337=0x1x8+x3x1026=0x2x914=0x3x10+x6x13+x7x1420=0x3x10+x4x11+x5x12+x6x13+x7x1440=0\begin{array}{rcl}x_{1} x_{8} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -30&=&0\\x_{1} x_{8} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -59&=&0\\x_{1} x_{8} +x_{3} x_{10} +2x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -48&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -37&=&0\\x_{1} x_{8} +x_{3} x_{10} -26&=&0\\x_{2} x_{9} -14&=&0\\x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -20&=&0\\x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -40&=&0\\\end{array}

A161A^{61}_1
h-characteristic: (1, 0, 0, 0, 1, 0, 1, 2)
Length of the weight dual to h: 122
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: D5+A1D^{1}_5+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V14ψ+2V11ψ+5V10ψ+4V9ψ+3V8ψ+2V7ψ+V6ψ+2V5ψ+4V4ψ+2V3ψ+2V2ψ+6Vψ+6V0V_{14\psi}+2V_{11\psi}+5V_{10\psi}+4V_{9\psi}+3V_{8\psi}+2V_{7\psi}+V_{6\psi}+2V_{5\psi}+4V_{4\psi}+2V_{3\psi}+2V_{2\psi}+6V_{\psi}+6V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=14h8+26h7+37h6+48h5+58h4+39h3+29h2+20h1e=10g51+8g48+g46+10g37+18g21+14g8f=g8+g21+g37+g46+g48+g51\begin{array}{rcl}h&=&14h_{8}+26h_{7}+37h_{6}+48h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\\ e&=&10g_{51}+8g_{48}+g_{46}+10g_{37}+18g_{21}+14g_{8}\\ f&=&g_{-8}+g_{-21}+g_{-37}+g_{-46}+g_{-48}+g_{-51}\end{array}
Lie brackets of the above elements.
h=14h8+26h7+37h6+48h5+58h4+39h3+29h2+20h1e=10g51+8g48+g46+10g37+18g21+14g8f=g8+g21+g37+g46+g48+g51\begin{array}{rcl}[e, f]&=&14h_{8}+26h_{7}+37h_{6}+48h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\\ [h, e]&=&20g_{51}+16g_{48}+2g_{46}+20g_{37}+36g_{21}+28g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-21}-2g_{-37}-2g_{-46}-2g_{-48}-2g_{-51}\end{array}
Centralizer type: A12+A1A^{2}_1+A_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 6): g2g_{-2}, h2h_{2}, h6h3h_{6}-h_{3}, g2g_{2}, g3+g6g_{3}+g_{-6}, g6+g3g_{6}+g_{-3}
Basis of centralizer intersected with cartan (dimension: 2): h2-h_{2}, h6h3h_{6}-h_{3}
Cartan of centralizer (dimension: 2): h2-h_{2}, h6h3h_{6}-h_{3}
Cartan-generating semisimple element: h6h3h2h_{6}-h_{3}-h_{2}
adjoint action: (200000000000000000000200000020000002)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
Characteristic polynomial ad H: x68x4+16x2x^6-8x^4+16x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -2)(x -2)(x +2)(x +2)
Eigenvalues of ad H: 00, 22, 2-2
6 eigenvectors of ad H: 0, 1, 0, 0, 0, 0(0,1,0,0,0,0), 0, 0, 1, 0, 0, 0(0,0,1,0,0,0), 1, 0, 0, 0, 0, 0(1,0,0,0,0,0), 0, 0, 0, 0, 0, 1(0,0,0,0,0,1), 0, 0, 0, 1, 0, 0(0,0,0,1,0,0), 0, 0, 0, 0, 1, 0(0,0,0,0,1,0)
Centralizer type: A^{2}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: (130)\begin{pmatrix}1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h2-h_{2}
matching e: g2g_{-2}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000200000000000000)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h2-h_{2}
matching e: g2g_{-2}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000200000000000000)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120)\begin{pmatrix}120\\ \end{pmatrix}

Scalar product computed: (160)\begin{pmatrix}1/60\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h6h3h_{6}-h_{3}
matching e: g6+g3g_{6}+g_{-3}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000000020000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h6h3h_{6}-h_{3}
matching e: g6+g3g_{6}+g_{-3}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000000020000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240)\begin{pmatrix}240\\ \end{pmatrix}
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Unknown elements.
h=14h8+26h7+37h6+48h5+58h4+39h3+29h2+20h1e=x4g51+x1g48+x6g46+x5g37+x3g21+x2g8e=x8g8+x9g21+x11g37+x12g46+x7g48+x10g51\begin{array}{rcl}h&=&14h_{8}+26h_{7}+37h_{6}+48h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\\ e&=&x_{4} g_{51}+x_{1} g_{48}+x_{6} g_{46}+x_{5} g_{37}+x_{3} g_{21}+x_{2} g_{8}\\ f&=&x_{8} g_{-8}+x_{9} g_{-21}+x_{11} g_{-37}+x_{12} g_{-46}+x_{7} g_{-48}+x_{10} g_{-51}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x814)h8+(x1x7+x3x926)h7+(x1x7+x3x9+x4x10+x6x1237)h6+(x1x7+x3x9+x4x10+x5x11+2x6x1248)h5+(2x1x7+2x4x10+2x5x11+2x6x1258)h4+(x1x7+2x4x10+x5x11+x6x1239)h3+(x1x7+x4x10+x5x11+x6x1229)h2+(x4x10+x5x1120)h1[e,f] - h = \left(x_{2} x_{8} -14\right)h_{8}+\left(x_{1} x_{7} +x_{3} x_{9} -26\right)h_{7}+\left(x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -37\right)h_{6}+\left(x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +2x_{6} x_{12} -48\right)h_{5}+\left(2x_{1} x_{7} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -58\right)h_{4}+\left(x_{1} x_{7} +2x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -39\right)h_{3}+\left(x_{1} x_{7} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -29\right)h_{2}+\left(x_{4} x_{10} +x_{5} x_{11} -20\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x7+x4x10+x5x11+x6x1229=0x1x7+2x4x10+x5x11+x6x1239=02x1x7+2x4x10+2x5x11+2x6x1258=0x1x7+x3x9+x4x10+x5x11+2x6x1248=0x1x7+x3x9+x4x10+x6x1237=0x1x7+x3x926=0x2x814=0x4x10+x5x1120=0\begin{array}{rcl}x_{1} x_{7} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -29&=&0\\x_{1} x_{7} +2x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -39&=&0\\2x_{1} x_{7} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -58&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +2x_{6} x_{12} -48&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -37&=&0\\x_{1} x_{7} +x_{3} x_{9} -26&=&0\\x_{2} x_{8} -14&=&0\\x_{4} x_{10} +x_{5} x_{11} -20&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=14h8+26h7+37h6+48h5+58h4+39h3+29h2+20h1e=x4g51+x1g48+x6g46+x5g37+x3g21+x2g8f=g8+g21+g37+g46+g48+g51\begin{array}{rcl}h&=&14h_{8}+26h_{7}+37h_{6}+48h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\\e&=&x_{4} g_{51}+x_{1} g_{48}+x_{6} g_{46}+x_{5} g_{37}+x_{3} g_{21}+x_{2} g_{8}\\f&=&g_{-8}+g_{-21}+g_{-37}+g_{-46}+g_{-48}+g_{-51}\end{array}
Matrix form of the system we are trying to solve: (100111100211200222101112101101101000010000000110)[col. vect.]=(2939584837261420)\begin{pmatrix}1 & 0 & 0 & 1 & 1 & 1\\ 1 & 0 & 0 & 2 & 1 & 1\\ 2 & 0 & 0 & 2 & 2 & 2\\ 1 & 0 & 1 & 1 & 1 & 2\\ 1 & 0 & 1 & 1 & 0 & 1\\ 1 & 0 & 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}29\\ 39\\ 58\\ 48\\ 37\\ 26\\ 14\\ 20\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=14h8+26h7+37h6+48h5+58h4+39h3+29h2+20h1e=x4g51+x1g48+x6g46+x5g37+x3g21+x2g8f=x8g8+x9g21+x11g37+x12g46+x7g48+x10g51\begin{array}{rcl}h&=&14h_{8}+26h_{7}+37h_{6}+48h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\\ e&=&x_{4} g_{51}+x_{1} g_{48}+x_{6} g_{46}+x_{5} g_{37}+x_{3} g_{21}+x_{2} g_{8}\\ f&=&x_{8} g_{-8}+x_{9} g_{-21}+x_{11} g_{-37}+x_{12} g_{-46}+x_{7} g_{-48}+x_{10} g_{-51}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x7+x4x10+x5x11+x6x1229=0x1x7+2x4x10+x5x11+x6x1239=02x1x7+2x4x10+2x5x11+2x6x1258=0x1x7+x3x9+x4x10+x5x11+2x6x1248=0x1x7+x3x9+x4x10+x6x1237=0x1x7+x3x926=0x2x814=0x4x10+x5x1120=0\begin{array}{rcl}x_{1} x_{7} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -29&=&0\\x_{1} x_{7} +2x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -39&=&0\\2x_{1} x_{7} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -58&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +2x_{6} x_{12} -48&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -37&=&0\\x_{1} x_{7} +x_{3} x_{9} -26&=&0\\x_{2} x_{8} -14&=&0\\x_{4} x_{10} +x_{5} x_{11} -20&=&0\\\end{array}

A160A^{60}_1
h-characteristic: (2, 0, 0, 0, 0, 0, 2, 2)
Length of the weight dual to h: 120
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: D5D^{1}_5
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V14ψ+9V10ψ+7V8ψ+V6ψ+8V4ψ+V2ψ+21V0V_{14\psi}+9V_{10\psi}+7V_{8\psi}+V_{6\psi}+8V_{4\psi}+V_{2\psi}+21V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=14h8+26h7+36h6+46h5+56h4+38h3+28h2+20h1e=10g69+8g60+10g32+14g8+18g7f=g7+g8+g32+g60+g69\begin{array}{rcl}h&=&14h_{8}+26h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\\ e&=&10g_{69}+8g_{60}+10g_{32}+14g_{8}+18g_{7}\\ f&=&g_{-7}+g_{-8}+g_{-32}+g_{-60}+g_{-69}\end{array}
Lie brackets of the above elements.
h=14h8+26h7+36h6+46h5+56h4+38h3+28h2+20h1e=10g69+8g60+10g32+14g8+18g7f=g7+g8+g32+g60+g69\begin{array}{rcl}[e, f]&=&14h_{8}+26h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\\ [h, e]&=&20g_{69}+16g_{60}+20g_{32}+28g_{8}+36g_{7}\\ [h, f]&=&-2g_{-7}-2g_{-8}-2g_{-32}-2g_{-60}-2g_{-69}\end{array}
Centralizer type: B3B_3
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 21): g19g_{-19}, g12g_{-12}, g11g_{-11}, g5g_{-5}, g4g_{-4}, g3g_{-3}, h3h_{3}, h4h_{4}, h5h_{5}, g2g31g_{2}-g_{-31}, g3g_{3}, g4g_{4}, g5g_{5}, g10g25g_{10}-g_{-25}, g11g_{11}, g12g_{12}, g17+g18g_{17}+g_{-18}, g18+g17g_{18}+g_{-17}, g19g_{19}, g25g10g_{25}-g_{-10}, g31g2g_{31}-g_{-2}
Basis of centralizer intersected with cartan (dimension: 3): h5-h_{5}, h4-h_{4}, h3-h_{3}
Cartan of centralizer (dimension: 3): h5-h_{5}, h4-h_{4}, h3-h_{3}
Cartan-generating semisimple element: h59h4+7h3-h_{5}-9h_{4}+7h_{3}
adjoint action: (60000000000000000000001700000000000000000000010000000000000000000007000000000000000000000240000000000000000000002300000000000000000000000000000000000000000000000000000000000000000000000000000000000000090000000000000000000002300000000000000000000024000000000000000000000700000000000000000000015000000000000000000000100000000000000000000017000000000000000000000800000000000000000000080000000000000000000006000000000000000000000150000000000000000000009)\begin{pmatrix}-6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 17 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 24 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -23 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 23 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -24 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -15 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -17 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 15 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9\\ \end{pmatrix}
Characteristic polynomial ad H: x211850x19+1331191x17477370000x15+92046550271x139719796344050x11+559177724467881x916437247604086500x7+197072635385350656x5181184937279897600x3x^{21}-1850x^{19}+1331191x^{17}-477370000x^{15}+92046550271x^{13}-9719796344050x^{11}+559177724467881x^9-16437247604086500x^7+197072635385350656x^5-181184937279897600x^3
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -24)(x -23)(x -17)(x -15)(x -9)(x -8)(x -7)(x -6)(x -1)(x +1)(x +6)(x +7)(x +8)(x +9)(x +15)(x +17)(x +23)(x +24)
Eigenvalues of ad H: 00, 2424, 2323, 1717, 1515, 99, 88, 77, 66, 11, 1-1, 6-6, 7-7, 8-8, 9-9, 15-15, 17-17, 23-23, 24-24
21 eigenvectors of ad H: 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)
Centralizer type: B^{1}_3
Reductive components (1 total):
Scalar product computed: (16016001601301600160130)\begin{pmatrix}1/60 & -1/60 & 0\\ -1/60 & 1/30 & -1/60\\ 0 & -1/60 & 1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (3 total):
h5+h3-h_{5}+h_{3}
matching e: g17+g18g_{17}+g_{-18}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000020000000000000000000002000000000000000000000200000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000200000000000000000000000000000000000000000002000000000000000000000200000000000000000000020000000000000000000002000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h4h3-h_{4}-h_{3}
matching e: g11g_{-11}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (100000000000000000000000000000000000000000002000000000000000000000100000000000000000000010000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000010000000000000000000001000000000000000000000100000000000000000000000000000000000000000002000000000000000000000000000000000000000000010000000000000000000001000000000000000000000100000000000000000000000000000000000000000001)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
h5+h4+h3h_{5}+h_{4}+h_{3}
matching e: g19g_{19}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000010000000000000000000001000000000000000000000100000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000010000000000000000000000000000000000000000000100000000000000000000010000000000000000000001000000000000000000000100000000000000000000000000000000000000000000000000000000000000000200000000000000000000010000000000000000000001)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h5+h3-h_{5}+h_{3}
matching e: g17+g18g_{17}+g_{-18}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000020000000000000000000002000000000000000000000200000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000200000000000000000000000000000000000000000002000000000000000000000200000000000000000000020000000000000000000002000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h4h3-h_{4}-h_{3}
matching e: g11g_{-11}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (100000000000000000000000000000000000000000002000000000000000000000100000000000000000000010000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000010000000000000000000001000000000000000000000100000000000000000000000000000000000000000002000000000000000000000000000000000000000000010000000000000000000001000000000000000000000100000000000000000000000000000000000000000001)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
h5+h4+h3h_{5}+h_{4}+h_{3}
matching e: g19g_{19}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000010000000000000000000001000000000000000000000100000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000010000000000000000000000000000000000000000000100000000000000000000010000000000000000000001000000000000000000000100000000000000000000000000000000000000000000000000000000000000000200000000000000000000010000000000000000000001)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
Elements in Cartan dual to root system: (1, 2, 1), (-1, -2, -1), (1, 1, 1), (-1, -1, -1), (1, 1, 0), (-1, -1, 0), (1, 2, 2), (-1, -2, -2), (1, 2, 0), (-1, -2, 0), (1, 0, 0), (-1, 0, 0), (0, 1, 1), (0, -1, -1), (0, 0, 1), (0, 0, -1), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240120012012060060120)\begin{pmatrix}240 & -120 & 0\\ -120 & 120 & -60\\ 0 & -60 & 120\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=14h8+26h7+36h6+46h5+56h4+38h3+28h2+20h1e=x4g69+x1g60+x5g32+x2g8+x3g7e=x8g7+x7g8+x10g32+x6g60+x9g69\begin{array}{rcl}h&=&14h_{8}+26h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\\ e&=&x_{4} g_{69}+x_{1} g_{60}+x_{5} g_{32}+x_{2} g_{8}+x_{3} g_{7}\\ f&=&x_{8} g_{-7}+x_{7} g_{-8}+x_{10} g_{-32}+x_{6} g_{-60}+x_{9} g_{-69}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x714)h8+(x1x6+x3x826)h7+(2x1x6+x4x9+x5x1036)h6+(2x1x6+2x4x9+x5x1046)h5+(2x1x6+3x4x9+x5x1056)h4+(x1x6+2x4x9+x5x1038)h3+(x1x6+2x4x928)h2+(x4x9+x5x1020)h1[e,f] - h = \left(x_{2} x_{7} -14\right)h_{8}+\left(x_{1} x_{6} +x_{3} x_{8} -26\right)h_{7}+\left(2x_{1} x_{6} +x_{4} x_{9} +x_{5} x_{10} -36\right)h_{6}+\left(2x_{1} x_{6} +2x_{4} x_{9} +x_{5} x_{10} -46\right)h_{5}+\left(2x_{1} x_{6} +3x_{4} x_{9} +x_{5} x_{10} -56\right)h_{4}+\left(x_{1} x_{6} +2x_{4} x_{9} +x_{5} x_{10} -38\right)h_{3}+\left(x_{1} x_{6} +2x_{4} x_{9} -28\right)h_{2}+\left(x_{4} x_{9} +x_{5} x_{10} -20\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x6+2x4x928=0x1x6+2x4x9+x5x1038=02x1x6+3x4x9+x5x1056=02x1x6+2x4x9+x5x1046=02x1x6+x4x9+x5x1036=0x1x6+x3x826=0x2x714=0x4x9+x5x1020=0\begin{array}{rcl}x_{1} x_{6} +2x_{4} x_{9} -28&=&0\\x_{1} x_{6} +2x_{4} x_{9} +x_{5} x_{10} -38&=&0\\2x_{1} x_{6} +3x_{4} x_{9} +x_{5} x_{10} -56&=&0\\2x_{1} x_{6} +2x_{4} x_{9} +x_{5} x_{10} -46&=&0\\2x_{1} x_{6} +x_{4} x_{9} +x_{5} x_{10} -36&=&0\\x_{1} x_{6} +x_{3} x_{8} -26&=&0\\x_{2} x_{7} -14&=&0\\x_{4} x_{9} +x_{5} x_{10} -20&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=14h8+26h7+36h6+46h5+56h4+38h3+28h2+20h1e=x4g69+x1g60+x5g32+x2g8+x3g7f=g7+g8+g32+g60+g69\begin{array}{rcl}h&=&14h_{8}+26h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\\e&=&x_{4} g_{69}+x_{1} g_{60}+x_{5} g_{32}+x_{2} g_{8}+x_{3} g_{7}\\f&=&g_{-7}+g_{-8}+g_{-32}+g_{-60}+g_{-69}\end{array}
Matrix form of the system we are trying to solve: (1002010021200312002120011101000100000011)[col. vect.]=(2838564636261420)\begin{pmatrix}1 & 0 & 0 & 2 & 0\\ 1 & 0 & 0 & 2 & 1\\ 2 & 0 & 0 & 3 & 1\\ 2 & 0 & 0 & 2 & 1\\ 2 & 0 & 0 & 1 & 1\\ 1 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}28\\ 38\\ 56\\ 46\\ 36\\ 26\\ 14\\ 20\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=14h8+26h7+36h6+46h5+56h4+38h3+28h2+20h1e=x4g69+x1g60+x5g32+x2g8+x3g7f=x8g7+x7g8+x10g32+x6g60+x9g69\begin{array}{rcl}h&=&14h_{8}+26h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\\ e&=&x_{4} g_{69}+x_{1} g_{60}+x_{5} g_{32}+x_{2} g_{8}+x_{3} g_{7}\\ f&=&x_{8} g_{-7}+x_{7} g_{-8}+x_{10} g_{-32}+x_{6} g_{-60}+x_{9} g_{-69}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x6+2x4x928=0x1x6+2x4x9+x5x1038=02x1x6+3x4x9+x5x1056=02x1x6+2x4x9+x5x1046=02x1x6+x4x9+x5x1036=0x1x6+x3x826=0x2x714=0x4x9+x5x1020=0\begin{array}{rcl}x_{1} x_{6} +2x_{4} x_{9} -28&=&0\\x_{1} x_{6} +2x_{4} x_{9} +x_{5} x_{10} -38&=&0\\2x_{1} x_{6} +3x_{4} x_{9} +x_{5} x_{10} -56&=&0\\2x_{1} x_{6} +2x_{4} x_{9} +x_{5} x_{10} -46&=&0\\2x_{1} x_{6} +x_{4} x_{9} +x_{5} x_{10} -36&=&0\\x_{1} x_{6} +x_{3} x_{8} -26&=&0\\x_{2} x_{7} -14&=&0\\x_{4} x_{9} +x_{5} x_{10} -20&=&0\\\end{array}

A157A^{57}_1
h-characteristic: (1, 0, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 114
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A6+A1A^{1}_6+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 3V12ψ+2V11ψ+V10ψ+2V9ψ+3V8ψ+4V7ψ+5V6ψ+4V5ψ+3V4ψ+2V3ψ+2V2ψ+2Vψ+3V03V_{12\psi}+2V_{11\psi}+V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+5V_{6\psi}+4V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=12h8+24h7+36h6+47h5+58h4+39h3+29h2+20h1e=6g50+6g36+12g34+g31+12g27+10g24+10g23f=g23+g24+g27+g31+g34+g36+g50\begin{array}{rcl}h&=&12h_{8}+24h_{7}+36h_{6}+47h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\\ e&=&6g_{50}+6g_{36}+12g_{34}+g_{31}+12g_{27}+10g_{24}+10g_{23}\\ f&=&g_{-23}+g_{-24}+g_{-27}+g_{-31}+g_{-34}+g_{-36}+g_{-50}\end{array}
Lie brackets of the above elements.
h=12h8+24h7+36h6+47h5+58h4+39h3+29h2+20h1e=6g50+6g36+12g34+g31+12g27+10g24+10g23f=g23+g24+g27+g31+g34+g36+g50\begin{array}{rcl}[e, f]&=&12h_{8}+24h_{7}+36h_{6}+47h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\\ [h, e]&=&12g_{50}+12g_{36}+24g_{34}+2g_{31}+24g_{27}+20g_{24}+20g_{23}\\ [h, f]&=&-2g_{-23}-2g_{-24}-2g_{-27}-2g_{-31}-2g_{-34}-2g_{-36}-2g_{-50}\end{array}
Centralizer type: A17A^{7}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 3): h712h5+12h312h2h_{7}-1/2h_{5}+1/2h_{3}-1/2h_{2}, g8+g5+g2+g3+2g15g_{8}+g_{5}+g_{2}+g_{-3}+2g_{-15}, g15+g3+g2+g5+2g8g_{15}+g_{3}+g_{-2}+g_{-5}+2g_{-8}
Basis of centralizer intersected with cartan (dimension: 1): 2h7h5+h3h22h_{7}-h_{5}+h_{3}-h_{2}
Cartan of centralizer (dimension: 1): 2h7h5+h3h22h_{7}-h_{5}+h_{3}-h_{2}
Cartan-generating semisimple element: 2h7h5+h3h22h_{7}-h_{5}+h_{3}-h_{2}
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Characteristic polynomial ad H: x34xx^3-4x
Factorization of characteristic polynomial of ad H: (x )(x -2)(x +2)
Eigenvalues of ad H: 00, 22, 2-2
3 eigenvectors of ad H: 1, 0, 0(1,0,0), 0, 0, 1(0,0,1), 0, 1, 0(0,1,0)
Centralizer type: A^{7}_1
Reductive components (1 total):
Scalar product computed: (1210)\begin{pmatrix}1/210\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
2h7h5+h3h22h_{7}-h_{5}+h_{3}-h_{2}
matching e: g15+g3+g2+g5+2g8g_{15}+g_{3}+g_{-2}+g_{-5}+2g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
2h7h5+h3h22h_{7}-h_{5}+h_{3}-h_{2}
matching e: g15+g3+g2+g5+2g8g_{15}+g_{3}+g_{-2}+g_{-5}+2g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (840)\begin{pmatrix}840\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=12h8+24h7+36h6+47h5+58h4+39h3+29h2+20h1e=x1g50+x6g36+x3g34+x7g31+x4g27+x2g24+x5g23e=x12g23+x9g24+x11g27+x14g31+x10g34+x13g36+x8g50\begin{array}{rcl}h&=&12h_{8}+24h_{7}+36h_{6}+47h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\\ e&=&x_{1} g_{50}+x_{6} g_{36}+x_{3} g_{34}+x_{7} g_{31}+x_{4} g_{27}+x_{2} g_{24}+x_{5} g_{23}\\ f&=&x_{12} g_{-23}+x_{9} g_{-24}+x_{11} g_{-27}+x_{14} g_{-31}+x_{10} g_{-34}+x_{13} g_{-36}+x_{8} g_{-50}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x8+x6x1312)h8+(x1x8+x3x10+x6x1324)h7+(x1x8+x3x10+x4x11+x6x1336)h6+(x1x8+x2x9+x3x10+x4x11+x6x13+x7x1447)h5+(x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1458)h4+(x1x8+x2x9+x4x11+x5x12+x7x1439)h3+(x1x8+x3x10+x5x12+x7x1429)h2+(x2x9+x5x1220)h1[e,f] - h = \left(x_{1} x_{8} +x_{6} x_{13} -12\right)h_{8}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} -24\right)h_{7}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -36\right)h_{6}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -47\right)h_{5}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -58\right)h_{4}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -39\right)h_{3}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -29\right)h_{2}+\left(x_{2} x_{9} +x_{5} x_{12} -20\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x3x10+x5x12+x7x1429=0x1x8+x2x9+x4x11+x5x12+x7x1439=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1458=0x1x8+x2x9+x3x10+x4x11+x6x13+x7x1447=0x1x8+x3x10+x4x11+x6x1336=0x1x8+x3x10+x6x1324=0x1x8+x6x1312=0x2x9+x5x1220=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -29&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -39&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -58&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -47&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -36&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} -24&=&0\\x_{1} x_{8} +x_{6} x_{13} -12&=&0\\x_{2} x_{9} +x_{5} x_{12} -20&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=12h8+24h7+36h6+47h5+58h4+39h3+29h2+20h1e=x1g50+x6g36+x3g34+x7g31+x4g27+x2g24+x5g23f=g23+g24+g27+g31+g34+g36+g50\begin{array}{rcl}h&=&12h_{8}+24h_{7}+36h_{6}+47h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\\e&=&x_{1} g_{50}+x_{6} g_{36}+x_{3} g_{34}+x_{7} g_{31}+x_{4} g_{27}+x_{2} g_{24}+x_{5} g_{23}\\f&=&g_{-23}+g_{-24}+g_{-27}+g_{-31}+g_{-34}+g_{-36}+g_{-50}\end{array}
Matrix form of the system we are trying to solve: (10101011101101111111211110111011010101001010000100100100)[col. vect.]=(2939584736241220)\begin{pmatrix}1 & 0 & 1 & 0 & 1 & 0 & 1\\ 1 & 1 & 0 & 1 & 1 & 0 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 2\\ 1 & 1 & 1 & 1 & 0 & 1 & 1\\ 1 & 0 & 1 & 1 & 0 & 1 & 0\\ 1 & 0 & 1 & 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}29\\ 39\\ 58\\ 47\\ 36\\ 24\\ 12\\ 20\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=12h8+24h7+36h6+47h5+58h4+39h3+29h2+20h1e=x1g50+x6g36+x3g34+x7g31+x4g27+x2g24+x5g23f=x12g23+x9g24+x11g27+x14g31+x10g34+x13g36+x8g50\begin{array}{rcl}h&=&12h_{8}+24h_{7}+36h_{6}+47h_{5}+58h_{4}+39h_{3}+29h_{2}+20h_{1}\\ e&=&x_{1} g_{50}+x_{6} g_{36}+x_{3} g_{34}+x_{7} g_{31}+x_{4} g_{27}+x_{2} g_{24}+x_{5} g_{23}\\ f&=&x_{12} g_{-23}+x_{9} g_{-24}+x_{11} g_{-27}+x_{14} g_{-31}+x_{10} g_{-34}+x_{13} g_{-36}+x_{8} g_{-50}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x3x10+x5x12+x7x1429=0x1x8+x2x9+x4x11+x5x12+x7x1439=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1458=0x1x8+x2x9+x3x10+x4x11+x6x13+x7x1447=0x1x8+x3x10+x4x11+x6x1336=0x1x8+x3x10+x6x1324=0x1x8+x6x1312=0x2x9+x5x1220=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -29&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -39&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -58&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -47&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -36&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} -24&=&0\\x_{1} x_{8} +x_{6} x_{13} -12&=&0\\x_{2} x_{9} +x_{5} x_{12} -20&=&0\\\end{array}

A156A^{56}_1
h-characteristic: (2, 0, 0, 0, 0, 2, 0, 0)
Length of the weight dual to h: 112
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 2D42D^{1}_4 Containing regular semisimple subalgebra number 2: A6A^{1}_6
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 3V12ψ+5V10ψ+3V8ψ+13V6ψ+3V4ψ+5V2ψ+6V03V_{12\psi}+5V_{10\psi}+3V_{8\psi}+13V_{6\psi}+3V_{4\psi}+5V_{2\psi}+6V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=12h8+24h7+36h6+46h5+56h4+38h3+28h2+20h1e=6g62+6g35+6g34+6g33+10g24+10g23+6g22+6g20f=g20+g22+g23+g24+g33+g34+g35+g62\begin{array}{rcl}h&=&12h_{8}+24h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\\ e&=&6g_{62}+6g_{35}+6g_{34}+6g_{33}+10g_{24}+10g_{23}+6g_{22}+6g_{20}\\ f&=&g_{-20}+g_{-22}+g_{-23}+g_{-24}+g_{-33}+g_{-34}+g_{-35}+g_{-62}\end{array}
Lie brackets of the above elements.
h=12h8+24h7+36h6+46h5+56h4+38h3+28h2+20h1e=6g62+6g35+6g34+6g33+10g24+10g23+6g22+6g20f=g20+g22+g23+g24+g33+g34+g35+g62\begin{array}{rcl}[e, f]&=&12h_{8}+24h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\\ [h, e]&=&12g_{62}+12g_{35}+12g_{34}+12g_{33}+20g_{24}+20g_{23}+12g_{22}+12g_{20}\\ [h, f]&=&-2g_{-20}-2g_{-22}-2g_{-23}-2g_{-24}-2g_{-33}-2g_{-34}-2g_{-35}-2g_{-62}\end{array}
Centralizer type: A17+A1A^{7}_1+A_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 6): g7+g3+g3+g7g_{7}+g_{3}+g_{-3}+g_{-7}, g15+g12+g11g10+g10g11g12g15g_{15}+g_{12}+g_{11}-g_{10}+g_{-10}-g_{-11}-g_{-12}-g_{-15}, g18+g8+g8+g18g_{18}+g_{8}+g_{-8}+g_{-18}, g19g17+g8+g4+g4+g8g17+g19g_{19}-g_{17}+g_{8}+g_{4}+g_{-4}+g_{-8}-g_{-17}+g_{-19}, g25g12g11+g10g10+g11+g12g25g_{25}-g_{12}-g_{11}+g_{10}-g_{-10}+g_{-11}+g_{-12}-g_{-25}, g31g5+g3g2g2+g3g5+g31g_{31}-g_{5}+g_{3}-g_{2}-g_{-2}+g_{-3}-g_{-5}+g_{-31}
Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 2): 72g3132g252g19+2g17g15+12g12+12g1112g102g8g7+72g52g492g3+72g2+72g292g32g4+72g5g72g8+12g1012g1112g12+g15+2g172g19+32g2572g31-7/2g_{31}-3/2g_{25}-2g_{19}+2g_{17}-g_{15}+1/2g_{12}+1/2g_{11}-1/2g_{10}-2g_{8}-g_{7}+7/2g_{5}-2g_{4}-9/2g_{3}+7/2g_{2}+7/2g_{-2}-9/2g_{-3}-2g_{-4}+7/2g_{-5}-g_{-7}-2g_{-8}+1/2g_{-10}-1/2g_{-11}-1/2g_{-12}+g_{-15}+2g_{-17}-2g_{-19}+3/2g_{-25}-7/2g_{-31}, 52g31+12g25+g19g18g1712g1212g11+12g1052g5+g4+52g352g252g2+52g3+g452g512g10+12g11+12g12g17g18+g1912g25+52g315/2g_{31}+1/2g_{25}+g_{19}-g_{18}-g_{17}-1/2g_{12}-1/2g_{11}+1/2g_{10}-5/2g_{5}+g_{4}+5/2g_{3}-5/2g_{2}-5/2g_{-2}+5/2g_{-3}+g_{-4}-5/2g_{-5}-1/2g_{-10}+1/2g_{-11}+1/2g_{-12}-g_{-17}-g_{-18}+g_{-19}-1/2g_{-25}+5/2g_{-31}
Cartan-generating semisimple element: g31+g25+g19+g18g17+g15+2g8+g7g5+g4+2g3g2g2+2g3+g4g5+g7+2g8g15g17+g18+g19g25+g31g_{31}+g_{25}+g_{19}+g_{18}-g_{17}+g_{15}+2g_{8}+g_{7}-g_{5}+g_{4}+2g_{3}-g_{2}-g_{-2}+2g_{-3}+g_{-4}-g_{-5}+g_{-7}+2g_{-8}-g_{-15}-g_{-17}+g_{-18}+g_{-19}-g_{-25}+g_{-31}
adjoint action: (021100201100130051040051102302010120)\begin{pmatrix}0 & -2 & 1 & 1 & 0 & 0\\ -2 & 0 & 1 & 1 & 0 & 0\\ -1 & -3 & 0 & 0 & 5 & -1\\ 0 & 4 & 0 & 0 & -5 & 1\\ -1 & 0 & 2 & -3 & 0 & 2\\ 0 & -1 & 0 & -1 & 2 & 0\\ \end{pmatrix}
Characteristic polynomial ad H: x632x4+112x2x^6-32x^4+112x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -2)(x +2)(x^2-28)
Eigenvalues of ad H: 00, 22, 2-2, 272\sqrt{7}, 27-2\sqrt{7}
6 eigenvectors of ad H: 5/4, 5/4, 7/4, 3/4, 1, 0(54,54,74,34,1,0), -1/4, -1/4, -3/4, 1/4, 0, 1(14,14,34,14,0,1), -4, 20/3, 4, 4/3, 5, 1(4,203,4,43,5,1), -4, -4, 0, 0, -3, 1(4,4,0,0,3,1), 0, 0, -2/3\sqrt{7}+10/3, 2/3\sqrt{7}-10/3, 4/3\sqrt{7}-5/3, 1(0,0,237+103,237103,43753,1), 0, 0, 2/3\sqrt{7}+10/3, -2/3\sqrt{7}-10/3, -4/3\sqrt{7}-5/3, 1(0,0,237+103,237103,43753,1)
Centralizer type: A^{7}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: (1210)\begin{pmatrix}1/210\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
14g31+34g25+12g19+32g1812g17+g15+14g12+14g1114g10+2g8+g7+14g5+12g4+34g3+14g2+14g2+34g3+12g4+14g5+g7+2g8+14g1014g1114g12g1512g17+32g18+12g1934g2514g31-1/4g_{31}+3/4g_{25}+1/2g_{19}+3/2g_{18}-1/2g_{17}+g_{15}+1/4g_{12}+1/4g_{11}-1/4g_{10}+2g_{8}+g_{7}+1/4g_{5}+1/2g_{4}+3/4g_{3}+1/4g_{2}+1/4g_{-2}+3/4g_{-3}+1/2g_{-4}+1/4g_{-5}+g_{-7}+2g_{-8}+1/4g_{-10}-1/4g_{-11}-1/4g_{-12}-g_{-15}-1/2g_{-17}+3/2g_{-18}+1/2g_{-19}-3/4g_{-25}-1/4g_{-31}
matching e: g31+5g25+43g19+4g1843g17+203g15+53g12+53g1153g10+163g84g7g5+43g43g3g2g23g3+43g4g54g7+163g8+53g1053g1153g12203g1543g17+4g18+43g195g25+g31g_{31}+5g_{25}+4/3g_{19}+4g_{18}-4/3g_{17}+20/3g_{15}+5/3g_{12}+5/3g_{11}-5/3g_{10}+16/3g_{8}-4g_{7}-g_{5}+4/3g_{4}-3g_{3}-g_{2}-g_{-2}-3g_{-3}+4/3g_{-4}-g_{-5}-4g_{-7}+16/3g_{-8}+5/3g_{-10}-5/3g_{-11}-5/3g_{-12}-20/3g_{-15}-4/3g_{-17}+4g_{-18}+4/3g_{-19}-5g_{-25}+g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0211002011003434000014140000320343400012141400)\begin{pmatrix}0 & -2 & 1 & 1 & 0 & 0\\ -2 & 0 & 1 & 1 & 0 & 0\\ -3/4 & 3/4 & 0 & 0 & 0 & 0\\ -1/4 & 1/4 & 0 & 0 & 0 & 0\\ -3/2 & 0 & 3/4 & 3/4 & 0 & 0\\ 0 & 1/2 & -1/4 & -1/4 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
14g31+34g25+12g19+32g1812g17+g15+14g12+14g1114g10+2g8+g7+14g5+12g4+34g3+14g2+14g2+34g3+12g4+14g5+g7+2g8+14g1014g1114g12g1512g17+32g18+12g1934g2514g31-1/4g_{31}+3/4g_{25}+1/2g_{19}+3/2g_{18}-1/2g_{17}+g_{15}+1/4g_{12}+1/4g_{11}-1/4g_{10}+2g_{8}+g_{7}+1/4g_{5}+1/2g_{4}+3/4g_{3}+1/4g_{2}+1/4g_{-2}+3/4g_{-3}+1/2g_{-4}+1/4g_{-5}+g_{-7}+2g_{-8}+1/4g_{-10}-1/4g_{-11}-1/4g_{-12}-g_{-15}-1/2g_{-17}+3/2g_{-18}+1/2g_{-19}-3/4g_{-25}-1/4g_{-31}
matching e: g31+5g25+43g19+4g1843g17+203g15+53g12+53g1153g10+163g84g7g5+43g43g3g2g23g3+43g4g54g7+163g8+53g1053g1153g12203g1543g17+4g18+43g195g25+g31g_{31}+5g_{25}+4/3g_{19}+4g_{18}-4/3g_{17}+20/3g_{15}+5/3g_{12}+5/3g_{11}-5/3g_{10}+16/3g_{8}-4g_{7}-g_{5}+4/3g_{4}-3g_{3}-g_{2}-g_{-2}-3g_{-3}+4/3g_{-4}-g_{-5}-4g_{-7}+16/3g_{-8}+5/3g_{-10}-5/3g_{-11}-5/3g_{-12}-20/3g_{-15}-4/3g_{-17}+4g_{-18}+4/3g_{-19}-5g_{-25}+g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0211002011003434000014140000320343400012141400)\begin{pmatrix}0 & -2 & 1 & 1 & 0 & 0\\ -2 & 0 & 1 & 1 & 0 & 0\\ -3/4 & 3/4 & 0 & 0 & 0 & 0\\ -1/4 & 1/4 & 0 & 0 & 0 & 0\\ -3/2 & 0 & 3/4 & 3/4 & 0 & 0\\ 0 & 1/2 & -1/4 & -1/4 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (840)\begin{pmatrix}840\\ \end{pmatrix}

Scalar product computed: (130)\begin{pmatrix}1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
5287g31+1287g25+1147g191147g181147g171287g121287g11+1287g105287g5+1147g4+5287g35287g25287g2+5287g3+1147g45287g51287g10+1287g11+1287g121147g171147g18+1147g191287g25+5287g315/28\sqrt{7}g_{31}+1/28\sqrt{7}g_{25}+1/14\sqrt{7}g_{19}-1/14\sqrt{7}g_{18}-1/14\sqrt{7}g_{17}-1/28\sqrt{7}g_{12}-1/28\sqrt{7}g_{11}+1/28\sqrt{7}g_{10}-5/28\sqrt{7}g_{5}+1/14\sqrt{7}g_{4}+5/28\sqrt{7}g_{3}-5/28\sqrt{7}g_{2}-5/28\sqrt{7}g_{-2}+5/28\sqrt{7}g_{-3}+1/14\sqrt{7}g_{-4}-5/28\sqrt{7}g_{-5}-1/28\sqrt{7}g_{-10}+1/28\sqrt{7}g_{-11}+1/28\sqrt{7}g_{-12}-1/14\sqrt{7}g_{-17}-1/14\sqrt{7}g_{-18}+1/14\sqrt{7}g_{-19}-1/28\sqrt{7}g_{-25}+5/28\sqrt{7}g_{-31}
matching e: g31+(43753)g25+(237103)g19+(237+103)g18+(237+103)g17+(437+53)g12+(437+53)g11+(43753)g10g5+(237103)g4+g3g2g2+g3+(237103)g4g5+(437+53)g10+(43753)g11+(43753)g12+(237+103)g17+(237+103)g18+(237103)g19+(437+53)g25+g31g_{31}+\left(4/3\sqrt{7}-5/3\right)g_{25}+\left(2/3\sqrt{7}-10/3\right)g_{19}+\left(-2/3\sqrt{7}+10/3\right)g_{18}+\left(-2/3\sqrt{7}+10/3\right)g_{17}+\left(-4/3\sqrt{7}+5/3\right)g_{12}+\left(-4/3\sqrt{7}+5/3\right)g_{11}+\left(4/3\sqrt{7}-5/3\right)g_{10}-g_{5}+\left(2/3\sqrt{7}-10/3\right)g_{4}+g_{3}-g_{2}-g_{-2}+g_{-3}+\left(2/3\sqrt{7}-10/3\right)g_{-4}-g_{-5}+\left(-4/3\sqrt{7}+5/3\right)g_{-10}+\left(4/3\sqrt{7}-5/3\right)g_{-11}+\left(4/3\sqrt{7}-5/3\right)g_{-12}+\left(-2/3\sqrt{7}+10/3\right)g_{-17}+\left(-2/3\sqrt{7}+10/3\right)g_{-18}+\left(2/3\sqrt{7}-10/3\right)g_{-19}+\left(-4/3\sqrt{7}+5/3\right)g_{-25}+g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000128715287005771771287152870057717711470528715287027703147128732872770)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ -1/28\sqrt{7} & -15/28\sqrt{7} & 0 & 0 & 5/7\sqrt{7} & -1/7\sqrt{7}\\ 1/28\sqrt{7} & 15/28\sqrt{7} & 0 & 0 & -5/7\sqrt{7} & 1/7\sqrt{7}\\ 1/14\sqrt{7} & 0 & 5/28\sqrt{7} & -15/28\sqrt{7} & 0 & 2/7\sqrt{7}\\ 0 & -3/14\sqrt{7} & 1/28\sqrt{7} & -3/28\sqrt{7} & 2/7\sqrt{7} & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
5287g31+1287g25+1147g191147g181147g171287g121287g11+1287g105287g5+1147g4+5287g35287g25287g2+5287g3+1147g45287g51287g10+1287g11+1287g121147g171147g18+1147g191287g25+5287g315/28\sqrt{7}g_{31}+1/28\sqrt{7}g_{25}+1/14\sqrt{7}g_{19}-1/14\sqrt{7}g_{18}-1/14\sqrt{7}g_{17}-1/28\sqrt{7}g_{12}-1/28\sqrt{7}g_{11}+1/28\sqrt{7}g_{10}-5/28\sqrt{7}g_{5}+1/14\sqrt{7}g_{4}+5/28\sqrt{7}g_{3}-5/28\sqrt{7}g_{2}-5/28\sqrt{7}g_{-2}+5/28\sqrt{7}g_{-3}+1/14\sqrt{7}g_{-4}-5/28\sqrt{7}g_{-5}-1/28\sqrt{7}g_{-10}+1/28\sqrt{7}g_{-11}+1/28\sqrt{7}g_{-12}-1/14\sqrt{7}g_{-17}-1/14\sqrt{7}g_{-18}+1/14\sqrt{7}g_{-19}-1/28\sqrt{7}g_{-25}+5/28\sqrt{7}g_{-31}
matching e: g31+(43753)g25+(237103)g19+(237+103)g18+(237+103)g17+(437+53)g12+(437+53)g11+(43753)g10g5+(237103)g4+g3g2g2+g3+(237103)g4g5+(437+53)g10+(43753)g11+(43753)g12+(237+103)g17+(237+103)g18+(237103)g19+(437+53)g25+g31g_{31}+\left(4/3\sqrt{7}-5/3\right)g_{25}+\left(2/3\sqrt{7}-10/3\right)g_{19}+\left(-2/3\sqrt{7}+10/3\right)g_{18}+\left(-2/3\sqrt{7}+10/3\right)g_{17}+\left(-4/3\sqrt{7}+5/3\right)g_{12}+\left(-4/3\sqrt{7}+5/3\right)g_{11}+\left(4/3\sqrt{7}-5/3\right)g_{10}-g_{5}+\left(2/3\sqrt{7}-10/3\right)g_{4}+g_{3}-g_{2}-g_{-2}+g_{-3}+\left(2/3\sqrt{7}-10/3\right)g_{-4}-g_{-5}+\left(-4/3\sqrt{7}+5/3\right)g_{-10}+\left(4/3\sqrt{7}-5/3\right)g_{-11}+\left(4/3\sqrt{7}-5/3\right)g_{-12}+\left(-2/3\sqrt{7}+10/3\right)g_{-17}+\left(-2/3\sqrt{7}+10/3\right)g_{-18}+\left(2/3\sqrt{7}-10/3\right)g_{-19}+\left(-4/3\sqrt{7}+5/3\right)g_{-25}+g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000128715287005771771287152870057717711470528715287027703147128732872770)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ -1/28\sqrt{7} & -15/28\sqrt{7} & 0 & 0 & 5/7\sqrt{7} & -1/7\sqrt{7}\\ 1/28\sqrt{7} & 15/28\sqrt{7} & 0 & 0 & -5/7\sqrt{7} & 1/7\sqrt{7}\\ 1/14\sqrt{7} & 0 & 5/28\sqrt{7} & -15/28\sqrt{7} & 0 & 2/7\sqrt{7}\\ 0 & -3/14\sqrt{7} & 1/28\sqrt{7} & -3/28\sqrt{7} & 2/7\sqrt{7} & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120)\begin{pmatrix}120\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=12h8+24h7+36h6+46h5+56h4+38h3+28h2+20h1e=x1g62+x3g35+x5g34+x7g33+x6g24+x2g23+x8g22+x4g20e=x12g20+x16g22+x10g23+x14g24+x15g33+x13g34+x11g35+x9g62\begin{array}{rcl}h&=&12h_{8}+24h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\\ e&=&x_{1} g_{62}+x_{3} g_{35}+x_{5} g_{34}+x_{7} g_{33}+x_{6} g_{24}+x_{2} g_{23}+x_{8} g_{22}+x_{4} g_{20}\\ f&=&x_{12} g_{-20}+x_{16} g_{-22}+x_{10} g_{-23}+x_{14} g_{-24}+x_{15} g_{-33}+x_{13} g_{-34}+x_{11} g_{-35}+x_{9} g_{-62}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x9+x8x1612)h8+(x1x9+x3x11+x5x13+x8x1624)h7+(x1x9+x3x11+x4x12+x5x13+x7x15+x8x1636)h6+(2x1x9+x3x11+x4x12+x5x13+x6x14+x7x1546)h5+(2x1x9+x2x10+x3x11+x4x12+x5x13+x6x14+x7x1556)h4+(x1x9+x2x10+x3x11+x6x14+x7x1538)h3+(x1x9+x2x10+x5x13+x7x1528)h2+(x2x10+x6x1420)h1[e,f] - h = \left(x_{1} x_{9} +x_{8} x_{16} -12\right)h_{8}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{8} x_{16} -24\right)h_{7}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -36\right)h_{6}+\left(2x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} -46\right)h_{5}+\left(2x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} -56\right)h_{4}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{6} x_{14} +x_{7} x_{15} -38\right)h_{3}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{7} x_{15} -28\right)h_{2}+\left(x_{2} x_{10} +x_{6} x_{14} -20\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x9+x2x10+x5x13+x7x1528=0x1x9+x2x10+x3x11+x6x14+x7x1538=02x1x9+x2x10+x3x11+x4x12+x5x13+x6x14+x7x1556=02x1x9+x3x11+x4x12+x5x13+x6x14+x7x1546=0x1x9+x3x11+x4x12+x5x13+x7x15+x8x1636=0x1x9+x3x11+x5x13+x8x1624=0x1x9+x8x1612=0x2x10+x6x1420=0\begin{array}{rcl}x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{7} x_{15} -28&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{6} x_{14} +x_{7} x_{15} -38&=&0\\2x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} -56&=&0\\2x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} -46&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -36&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{8} x_{16} -24&=&0\\x_{1} x_{9} +x_{8} x_{16} -12&=&0\\x_{2} x_{10} +x_{6} x_{14} -20&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=12h8+24h7+36h6+46h5+56h4+38h3+28h2+20h1e=x1g62+x3g35+x5g34+x7g33+x6g24+x2g23+x8g22+x4g20f=g20+g22+g23+g24+g33+g34+g35+g62\begin{array}{rcl}h&=&12h_{8}+24h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\\e&=&x_{1} g_{62}+x_{3} g_{35}+x_{5} g_{34}+x_{7} g_{33}+x_{6} g_{24}+x_{2} g_{23}+x_{8} g_{22}+x_{4} g_{20}\\f&=&g_{-20}+g_{-22}+g_{-23}+g_{-24}+g_{-33}+g_{-34}+g_{-35}+g_{-62}\end{array}
Matrix form of the system we are trying to solve: (1100101011100110211111102011111010111011101010011000000101000100)[col. vect.]=(2838564636241220)\begin{pmatrix}1 & 1 & 0 & 0 & 1 & 0 & 1 & 0\\ 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0\\ 2 & 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 2 & 0 & 1 & 1 & 1 & 1 & 1 & 0\\ 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1\\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}28\\ 38\\ 56\\ 46\\ 36\\ 24\\ 12\\ 20\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=12h8+24h7+36h6+46h5+56h4+38h3+28h2+20h1e=x1g62+x3g35+x5g34+x7g33+x6g24+x2g23+x8g22+x4g20f=x12g20+x16g22+x10g23+x14g24+x15g33+x13g34+x11g35+x9g62\begin{array}{rcl}h&=&12h_{8}+24h_{7}+36h_{6}+46h_{5}+56h_{4}+38h_{3}+28h_{2}+20h_{1}\\ e&=&x_{1} g_{62}+x_{3} g_{35}+x_{5} g_{34}+x_{7} g_{33}+x_{6} g_{24}+x_{2} g_{23}+x_{8} g_{22}+x_{4} g_{20}\\ f&=&x_{12} g_{-20}+x_{16} g_{-22}+x_{10} g_{-23}+x_{14} g_{-24}+x_{15} g_{-33}+x_{13} g_{-34}+x_{11} g_{-35}+x_{9} g_{-62}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x9+x2x10+x5x13+x7x1528=0x1x9+x2x10+x3x11+x6x14+x7x1538=02x1x9+x2x10+x3x11+x4x12+x5x13+x6x14+x7x1556=02x1x9+x3x11+x4x12+x5x13+x6x14+x7x1546=0x1x9+x3x11+x4x12+x5x13+x7x15+x8x1636=0x1x9+x3x11+x5x13+x8x1624=0x1x9+x8x1612=0x2x10+x6x1420=0\begin{array}{rcl}x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{7} x_{15} -28&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{6} x_{14} +x_{7} x_{15} -38&=&0\\2x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} -56&=&0\\2x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} -46&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{7} x_{15} +x_{8} x_{16} -36&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{8} x_{16} -24&=&0\\x_{1} x_{9} +x_{8} x_{16} -12&=&0\\x_{2} x_{10} +x_{6} x_{14} -20&=&0\\\end{array}

A140A^{40}_1
h-characteristic: (0, 0, 0, 0, 2, 0, 0, 0)
Length of the weight dual to h: 80
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 9
Containing regular semisimple subalgebra number 1: A5+A2+A1A^{1}_5+A^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 2: 2A42A^{1}_4 Containing regular semisimple subalgebra number 3: E8E^{1}_8 Containing regular semisimple subalgebra number 4: D8D^{1}_8 Containing regular semisimple subalgebra number 5: E7+A1E^{1}_7+A^{1}_1 Containing regular semisimple subalgebra number 6: E6+A2E^{1}_6+A^{1}_2 Containing regular semisimple subalgebra number 7: D6+2A1D^{1}_6+2A^{1}_1 Containing regular semisimple subalgebra number 8: D5+A3D^{1}_5+A^{1}_3 Containing regular semisimple subalgebra number 9: 2D42D^{1}_4
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 4V10ψ+6V8ψ+10V6ψ+10V4ψ+10V2ψ4V_{10\psi}+6V_{8\psi}+10V_{6\psi}+10V_{4\psi}+10V_{2\psi}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+20h7+30h6+40h5+48h4+32h3+24h2+16h1e=5g66+2g35+8g34+g33+9g32+8g31+2g30+5g29f=g29+g30+g31+g32+g33+g34+g35+g66\begin{array}{rcl}h&=&10h_{8}+20h_{7}+30h_{6}+40h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\\ e&=&5g_{66}+2g_{35}+8g_{34}+g_{33}+9g_{32}+8g_{31}+2g_{30}+5g_{29}\\ f&=&g_{-29}+g_{-30}+g_{-31}+g_{-32}+g_{-33}+g_{-34}+g_{-35}+g_{-66}\end{array}
Lie brackets of the above elements.
h=10h8+20h7+30h6+40h5+48h4+32h3+24h2+16h1e=5g66+2g35+8g34+g33+9g32+8g31+2g30+5g29f=g29+g30+g31+g32+g33+g34+g35+g66\begin{array}{rcl}[e, f]&=&10h_{8}+20h_{7}+30h_{6}+40h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\\ [h, e]&=&10g_{66}+4g_{35}+16g_{34}+2g_{33}+18g_{32}+16g_{31}+4g_{30}+10g_{29}\\ [h, f]&=&-2g_{-29}-2g_{-30}-2g_{-31}-2g_{-32}-2g_{-33}-2g_{-34}-2g_{-35}-2g_{-66}\end{array}
Centralizer type: 00
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Unknown elements.
h=10h8+20h7+30h6+40h5+48h4+32h3+24h2+16h1e=x1g66+x6g35+x2g34+x8g33+x3g32+x4g31+x7g30+x5g29e=x13g29+x15g30+x12g31+x11g32+x16g33+x10g34+x14g35+x9g66\begin{array}{rcl}h&=&10h_{8}+20h_{7}+30h_{6}+40h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\\ e&=&x_{1} g_{66}+x_{6} g_{35}+x_{2} g_{34}+x_{8} g_{33}+x_{3} g_{32}+x_{4} g_{31}+x_{7} g_{30}+x_{5} g_{29}\\ f&=&x_{13} g_{-29}+x_{15} g_{-30}+x_{12} g_{-31}+x_{11} g_{-32}+x_{16} g_{-33}+x_{10} g_{-34}+x_{14} g_{-35}+x_{9} g_{-66}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x9+x5x1310)h8+(x1x9+x2x10+x5x13+x6x1420)h7+(x1x9+x2x10+x3x11+x5x13+x6x14+x8x1630)h6+(x1x9+x2x10+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1640)h5+(2x1x9+x2x10+x3x11+2x4x12+x6x14+x7x15+x8x1648)h4+(2x1x9+x3x11+x4x12+x6x14+x7x15+x8x1632)h3+(x1x9+x2x10+x4x12+x7x15+x8x1624)h2+(x1x9+x3x11+x7x1516)h1[e,f] - h = \left(x_{1} x_{9} +x_{5} x_{13} -10\right)h_{8}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{6} x_{14} -20\right)h_{7}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{8} x_{16} -30\right)h_{6}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -40\right)h_{5}+\left(2x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +2x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -48\right)h_{4}+\left(2x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -32\right)h_{3}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{7} x_{15} +x_{8} x_{16} -24\right)h_{2}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{7} x_{15} -16\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x9+x3x11+x7x1516=0x1x9+x2x10+x4x12+x7x15+x8x1624=02x1x9+x3x11+x4x12+x6x14+x7x15+x8x1632=02x1x9+x2x10+x3x11+2x4x12+x6x14+x7x15+x8x1648=0x1x9+x2x10+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1640=0x1x9+x2x10+x3x11+x5x13+x6x14+x8x1630=0x1x9+x2x10+x5x13+x6x1420=0x1x9+x5x1310=0\begin{array}{rcl}x_{1} x_{9} +x_{3} x_{11} +x_{7} x_{15} -16&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{7} x_{15} +x_{8} x_{16} -24&=&0\\2x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -32&=&0\\2x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +2x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -48&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -40&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{8} x_{16} -30&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{6} x_{14} -20&=&0\\x_{1} x_{9} +x_{5} x_{13} -10&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+20h7+30h6+40h5+48h4+32h3+24h2+16h1e=x1g66+x6g35+x2g34+x8g33+x3g32+x4g31+x7g30+x5g29f=g29+g30+g31+g32+g33+g34+g35+g66\begin{array}{rcl}h&=&10h_{8}+20h_{7}+30h_{6}+40h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\\e&=&x_{1} g_{66}+x_{6} g_{35}+x_{2} g_{34}+x_{8} g_{33}+x_{3} g_{32}+x_{4} g_{31}+x_{7} g_{30}+x_{5} g_{29}\\f&=&g_{-29}+g_{-30}+g_{-31}+g_{-32}+g_{-33}+g_{-34}+g_{-35}+g_{-66}\end{array}
Matrix form of the system we are trying to solve: (1010001011010011201101112112011111111111111011011100110010001000)[col. vect.]=(1624324840302010)\begin{pmatrix}1 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\ 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1\\ 2 & 0 & 1 & 1 & 0 & 1 & 1 & 1\\ 2 & 1 & 1 & 2 & 0 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1\\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}16\\ 24\\ 32\\ 48\\ 40\\ 30\\ 20\\ 10\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+20h7+30h6+40h5+48h4+32h3+24h2+16h1e=x1g66+x6g35+x2g34+x8g33+x3g32+x4g31+x7g30+x5g29f=x13g29+x15g30+x12g31+x11g32+x16g33+x10g34+x14g35+x9g66\begin{array}{rcl}h&=&10h_{8}+20h_{7}+30h_{6}+40h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\\ e&=&x_{1} g_{66}+x_{6} g_{35}+x_{2} g_{34}+x_{8} g_{33}+x_{3} g_{32}+x_{4} g_{31}+x_{7} g_{30}+x_{5} g_{29}\\ f&=&x_{13} g_{-29}+x_{15} g_{-30}+x_{12} g_{-31}+x_{11} g_{-32}+x_{16} g_{-33}+x_{10} g_{-34}+x_{14} g_{-35}+x_{9} g_{-66}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x9+x3x11+x7x1516=0x1x9+x2x10+x4x12+x7x15+x8x1624=02x1x9+x3x11+x4x12+x6x14+x7x15+x8x1632=02x1x9+x2x10+x3x11+2x4x12+x6x14+x7x15+x8x1648=0x1x9+x2x10+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1640=0x1x9+x2x10+x3x11+x5x13+x6x14+x8x1630=0x1x9+x2x10+x5x13+x6x1420=0x1x9+x5x1310=0\begin{array}{rcl}x_{1} x_{9} +x_{3} x_{11} +x_{7} x_{15} -16&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{7} x_{15} +x_{8} x_{16} -24&=&0\\2x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -32&=&0\\2x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +2x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -48&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -40&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{8} x_{16} -30&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{6} x_{14} -20&=&0\\x_{1} x_{9} +x_{5} x_{13} -10&=&0\\\end{array}

A139A^{39}_1
h-characteristic: (0, 0, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 78
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: A5+A2A^{1}_5+A^{1}_2 Containing regular semisimple subalgebra number 2: E7E^{1}_7 Containing regular semisimple subalgebra number 3: D6+A1D^{1}_6+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 3V10ψ+2V9ψ+3V8ψ+4V7ψ+5V6ψ+6V5ψ+4V4ψ+6V3ψ+6V2ψ+3V03V_{10\psi}+2V_{9\psi}+3V_{8\psi}+4V_{7\psi}+5V_{6\psi}+6V_{5\psi}+4V_{4\psi}+6V_{3\psi}+6V_{2\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+20h7+30h6+39h5+48h4+32h3+24h2+16h1e=5g55+9g37+5g36+8g35+2g34+8g33+2g32f=g32+g33+g34+g35+g36+g37+g55\begin{array}{rcl}h&=&10h_{8}+20h_{7}+30h_{6}+39h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\\ e&=&5g_{55}+9g_{37}+5g_{36}+8g_{35}+2g_{34}+8g_{33}+2g_{32}\\ f&=&g_{-32}+g_{-33}+g_{-34}+g_{-35}+g_{-36}+g_{-37}+g_{-55}\end{array}
Lie brackets of the above elements.
h=10h8+20h7+30h6+39h5+48h4+32h3+24h2+16h1e=5g55+9g37+5g36+8g35+2g34+8g33+2g32f=g32+g33+g34+g35+g36+g37+g55\begin{array}{rcl}[e, f]&=&10h_{8}+20h_{7}+30h_{6}+39h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\\ [h, e]&=&10g_{55}+18g_{37}+10g_{36}+16g_{35}+4g_{34}+16g_{33}+4g_{32}\\ [h, f]&=&-2g_{-32}-2g_{-33}-2g_{-34}-2g_{-35}-2g_{-36}-2g_{-37}-2g_{-55}\end{array}
Centralizer type: A1A_1
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Unknown elements.
h=10h8+20h7+30h6+39h5+48h4+32h3+24h2+16h1e=x1g55+x3g37+x5g36+x2g35+x6g34+x4g33+x7g32e=x14g32+x11g33+x13g34+x9g35+x12g36+x10g37+x8g55\begin{array}{rcl}h&=&10h_{8}+20h_{7}+30h_{6}+39h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\\ e&=&x_{1} g_{55}+x_{3} g_{37}+x_{5} g_{36}+x_{2} g_{35}+x_{6} g_{34}+x_{4} g_{33}+x_{7} g_{32}\\ f&=&x_{14} g_{-32}+x_{11} g_{-33}+x_{13} g_{-34}+x_{9} g_{-35}+x_{12} g_{-36}+x_{10} g_{-37}+x_{8} g_{-55}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x8+x5x1210)h8+(x1x8+x2x9+x5x12+x6x1320)h7+(x1x8+x2x9+x4x11+x5x12+x6x13+x7x1430)h6+(x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+x7x1439)h5+(x1x8+x2x9+2x3x10+x4x11+x5x12+x6x13+x7x1448)h4+(x1x8+x2x9+x3x10+x4x11+x7x1432)h3+(x1x8+x3x10+x4x11+x6x1324)h2+(x1x8+x3x10+x7x1416)h1[e,f] - h = \left(x_{1} x_{8} +x_{5} x_{12} -10\right)h_{8}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{5} x_{12} +x_{6} x_{13} -20\right)h_{7}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -30\right)h_{6}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -39\right)h_{5}+\left(x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -48\right)h_{4}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{7} x_{14} -32\right)h_{3}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -24\right)h_{2}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{7} x_{14} -16\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x3x10+x7x1416=0x1x8+x3x10+x4x11+x6x1324=0x1x8+x2x9+x3x10+x4x11+x7x1432=0x1x8+x2x9+2x3x10+x4x11+x5x12+x6x13+x7x1448=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+x7x1439=0x1x8+x2x9+x4x11+x5x12+x6x13+x7x1430=0x1x8+x2x9+x5x12+x6x1320=0x1x8+x5x1210=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{7} x_{14} -16&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -24&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{7} x_{14} -32&=&0\\x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -48&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -39&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -30&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{5} x_{12} +x_{6} x_{13} -20&=&0\\x_{1} x_{8} +x_{5} x_{12} -10&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+20h7+30h6+39h5+48h4+32h3+24h2+16h1e=x1g55+x3g37+x5g36+x2g35+x6g34+x4g33+x7g32f=g32+g33+g34+g35+g36+g37+g55\begin{array}{rcl}h&=&10h_{8}+20h_{7}+30h_{6}+39h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\\e&=&x_{1} g_{55}+x_{3} g_{37}+x_{5} g_{36}+x_{2} g_{35}+x_{6} g_{34}+x_{4} g_{33}+x_{7} g_{32}\\f&=&g_{-32}+g_{-33}+g_{-34}+g_{-35}+g_{-36}+g_{-37}+g_{-55}\end{array}
Matrix form of the system we are trying to solve: (10100011011010111100111211111111111110111111001101000100)[col. vect.]=(1624324839302010)\begin{pmatrix}1 & 0 & 1 & 0 & 0 & 0 & 1\\ 1 & 0 & 1 & 1 & 0 & 1 & 0\\ 1 & 1 & 1 & 1 & 0 & 0 & 1\\ 1 & 1 & 2 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 0 & 1 & 1 & 1 & 1\\ 1 & 1 & 0 & 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}16\\ 24\\ 32\\ 48\\ 39\\ 30\\ 20\\ 10\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+20h7+30h6+39h5+48h4+32h3+24h2+16h1e=x1g55+x3g37+x5g36+x2g35+x6g34+x4g33+x7g32f=x14g32+x11g33+x13g34+x9g35+x12g36+x10g37+x8g55\begin{array}{rcl}h&=&10h_{8}+20h_{7}+30h_{6}+39h_{5}+48h_{4}+32h_{3}+24h_{2}+16h_{1}\\ e&=&x_{1} g_{55}+x_{3} g_{37}+x_{5} g_{36}+x_{2} g_{35}+x_{6} g_{34}+x_{4} g_{33}+x_{7} g_{32}\\ f&=&x_{14} g_{-32}+x_{11} g_{-33}+x_{13} g_{-34}+x_{9} g_{-35}+x_{12} g_{-36}+x_{10} g_{-37}+x_{8} g_{-55}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x3x10+x7x1416=0x1x8+x3x10+x4x11+x6x1324=0x1x8+x2x9+x3x10+x4x11+x7x1432=0x1x8+x2x9+2x3x10+x4x11+x5x12+x6x13+x7x1448=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+x7x1439=0x1x8+x2x9+x4x11+x5x12+x6x13+x7x1430=0x1x8+x2x9+x5x12+x6x1320=0x1x8+x5x1210=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{7} x_{14} -16&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -24&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{7} x_{14} -32&=&0\\x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -48&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -39&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -30&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{5} x_{12} +x_{6} x_{13} -20&=&0\\x_{1} x_{8} +x_{5} x_{12} -10&=&0\\\end{array}

A138A^{38}_1
h-characteristic: (0, 1, 1, 0, 0, 0, 1, 0)
Length of the weight dual to h: 76
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: D4+A3D^{1}_4+A^{1}_3 Containing regular semisimple subalgebra number 2: D6D^{1}_6
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 2V10ψ+4V9ψ+V8ψ+4V7ψ+7V6ψ+4V5ψ+5V4ψ+8V3ψ+3V2ψ+6V02V_{10\psi}+4V_{9\psi}+V_{8\psi}+4V_{7\psi}+7V_{6\psi}+4V_{5\psi}+5V_{4\psi}+8V_{3\psi}+3V_{2\psi}+6V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+20h7+29h6+38h5+47h4+32h3+24h2+16h1e=6g46+3g45+4g43+6g42+10g40+3g30+6g17f=g17+g30+g40+g42+g43+g45+g46\begin{array}{rcl}h&=&10h_{8}+20h_{7}+29h_{6}+38h_{5}+47h_{4}+32h_{3}+24h_{2}+16h_{1}\\ e&=&6g_{46}+3g_{45}+4g_{43}+6g_{42}+10g_{40}+3g_{30}+6g_{17}\\ f&=&g_{-17}+g_{-30}+g_{-40}+g_{-42}+g_{-43}+g_{-45}+g_{-46}\end{array}
Lie brackets of the above elements.
h=10h8+20h7+29h6+38h5+47h4+32h3+24h2+16h1e=6g46+3g45+4g43+6g42+10g40+3g30+6g17f=g17+g30+g40+g42+g43+g45+g46\begin{array}{rcl}[e, f]&=&10h_{8}+20h_{7}+29h_{6}+38h_{5}+47h_{4}+32h_{3}+24h_{2}+16h_{1}\\ [h, e]&=&12g_{46}+6g_{45}+8g_{43}+12g_{42}+20g_{40}+6g_{30}+12g_{17}\\ [h, f]&=&-2g_{-17}-2g_{-30}-2g_{-40}-2g_{-42}-2g_{-43}-2g_{-45}-2g_{-46}\end{array}
Centralizer type: 2A12A_1
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Unknown elements.
h=10h8+20h7+29h6+38h5+47h4+32h3+24h2+16h1e=x3g46+x5g45+x6g43+x1g42+x2g40+x7g30+x4g17e=x11g17+x14g30+x9g40+x8g42+x13g43+x12g45+x10g46\begin{array}{rcl}h&=&10h_{8}+20h_{7}+29h_{6}+38h_{5}+47h_{4}+32h_{3}+24h_{2}+16h_{1}\\ e&=&x_{3} g_{46}+x_{5} g_{45}+x_{6} g_{43}+x_{1} g_{42}+x_{2} g_{40}+x_{7} g_{30}+x_{4} g_{17}\\ f&=&x_{11} g_{-17}+x_{14} g_{-30}+x_{9} g_{-40}+x_{8} g_{-42}+x_{13} g_{-43}+x_{12} g_{-45}+x_{10} g_{-46}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x8+x6x1310)h8+(x1x8+x2x9+x6x1320)h7+(x1x8+x2x9+x3x10+x5x12+x6x1329)h6+(x1x8+x2x9+2x3x10+x5x12+x6x13+x7x1438)h5+(x1x8+x2x9+2x3x10+x4x11+2x5x12+x6x13+x7x1447)h4+(x2x9+x3x10+x4x11+x5x12+x6x13+x7x1432)h3+(x1x8+x3x10+x4x11+x5x12+x7x1424)h2+(x2x9+x5x12+x7x1416)h1[e,f] - h = \left(x_{1} x_{8} +x_{6} x_{13} -10\right)h_{8}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{6} x_{13} -20\right)h_{7}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -29\right)h_{6}+\left(x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -38\right)h_{5}+\left(x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -47\right)h_{4}+\left(x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -32\right)h_{3}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -24\right)h_{2}+\left(x_{2} x_{9} +x_{5} x_{12} +x_{7} x_{14} -16\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x3x10+x4x11+x5x12+x7x1424=0x1x8+x2x9+2x3x10+x4x11+2x5x12+x6x13+x7x1447=0x1x8+x2x9+2x3x10+x5x12+x6x13+x7x1438=0x1x8+x2x9+x3x10+x5x12+x6x1329=0x1x8+x2x9+x6x1320=0x1x8+x6x1310=0x2x9+x5x12+x7x1416=0x2x9+x3x10+x4x11+x5x12+x6x13+x7x1432=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -24&=&0\\x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -47&=&0\\x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -38&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -29&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{6} x_{13} -20&=&0\\x_{1} x_{8} +x_{6} x_{13} -10&=&0\\x_{2} x_{9} +x_{5} x_{12} +x_{7} x_{14} -16&=&0\\x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -32&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+20h7+29h6+38h5+47h4+32h3+24h2+16h1e=x3g46+x5g45+x6g43+x1g42+x2g40+x7g30+x4g17f=g17+g30+g40+g42+g43+g45+g46\begin{array}{rcl}h&=&10h_{8}+20h_{7}+29h_{6}+38h_{5}+47h_{4}+32h_{3}+24h_{2}+16h_{1}\\e&=&x_{3} g_{46}+x_{5} g_{45}+x_{6} g_{43}+x_{1} g_{42}+x_{2} g_{40}+x_{7} g_{30}+x_{4} g_{17}\\f&=&g_{-17}+g_{-30}+g_{-40}+g_{-42}+g_{-43}+g_{-45}+g_{-46}\end{array}
Matrix form of the system we are trying to solve: (10111011121211112011111101101100010100001001001010111111)[col. vect.]=(2447382920101632)\begin{pmatrix}1 & 0 & 1 & 1 & 1 & 0 & 1\\ 1 & 1 & 2 & 1 & 2 & 1 & 1\\ 1 & 1 & 2 & 0 & 1 & 1 & 1\\ 1 & 1 & 1 & 0 & 1 & 1 & 0\\ 1 & 1 & 0 & 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 1 & 1 & 1 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}24\\ 47\\ 38\\ 29\\ 20\\ 10\\ 16\\ 32\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+20h7+29h6+38h5+47h4+32h3+24h2+16h1e=x3g46+x5g45+x6g43+x1g42+x2g40+x7g30+x4g17f=x11g17+x14g30+x9g40+x8g42+x13g43+x12g45+x10g46\begin{array}{rcl}h&=&10h_{8}+20h_{7}+29h_{6}+38h_{5}+47h_{4}+32h_{3}+24h_{2}+16h_{1}\\ e&=&x_{3} g_{46}+x_{5} g_{45}+x_{6} g_{43}+x_{1} g_{42}+x_{2} g_{40}+x_{7} g_{30}+x_{4} g_{17}\\ f&=&x_{11} g_{-17}+x_{14} g_{-30}+x_{9} g_{-40}+x_{8} g_{-42}+x_{13} g_{-43}+x_{12} g_{-45}+x_{10} g_{-46}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x3x10+x4x11+x5x12+x7x1424=0x1x8+x2x9+2x3x10+x4x11+2x5x12+x6x13+x7x1447=0x1x8+x2x9+2x3x10+x5x12+x6x13+x7x1438=0x1x8+x2x9+x3x10+x5x12+x6x1329=0x1x8+x2x9+x6x1320=0x1x8+x6x1310=0x2x9+x5x12+x7x1416=0x2x9+x3x10+x4x11+x5x12+x6x13+x7x1432=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -24&=&0\\x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -47&=&0\\x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -38&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -29&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{6} x_{13} -20&=&0\\x_{1} x_{8} +x_{6} x_{13} -10&=&0\\x_{2} x_{9} +x_{5} x_{12} +x_{7} x_{14} -16&=&0\\x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -32&=&0\\\end{array}

A137A^{37}_1
h-characteristic: (1, 0, 0, 0, 1, 0, 1, 0)
Length of the weight dual to h: 74
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A5+2A1A^{1}_5+2A^{1}_1 Containing regular semisimple subalgebra number 2: E6+A1E^{1}_6+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 2V10ψ+2V9ψ+4V8ψ+4V7ψ+4V6ψ+6V5ψ+7V4ψ+4V3ψ+4V2ψ+4Vψ+3V02V_{10\psi}+2V_{9\psi}+4V_{8\psi}+4V_{7\psi}+4V_{6\psi}+6V_{5\psi}+7V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+20h7+29h6+38h5+46h4+31h3+23h2+16h1e=5g56+g46+8g38+8g37+9g35+g34+5g29f=g29+g34+g35+g37+g38+g46+g56\begin{array}{rcl}h&=&10h_{8}+20h_{7}+29h_{6}+38h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\\ e&=&5g_{56}+g_{46}+8g_{38}+8g_{37}+9g_{35}+g_{34}+5g_{29}\\ f&=&g_{-29}+g_{-34}+g_{-35}+g_{-37}+g_{-38}+g_{-46}+g_{-56}\end{array}
Lie brackets of the above elements.
h=10h8+20h7+29h6+38h5+46h4+31h3+23h2+16h1e=5g56+g46+8g38+8g37+9g35+g34+5g29f=g29+g34+g35+g37+g38+g46+g56\begin{array}{rcl}[e, f]&=&10h_{8}+20h_{7}+29h_{6}+38h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\\ [h, e]&=&10g_{56}+2g_{46}+16g_{38}+16g_{37}+18g_{35}+2g_{34}+10g_{29}\\ [h, f]&=&-2g_{-29}-2g_{-34}-2g_{-35}-2g_{-37}-2g_{-38}-2g_{-46}-2g_{-56}\end{array}
Centralizer type: A13A^{3}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 3): h6h3h2h_{6}-h_{3}-h_{2}, g6+g4g17g_{6}+g_{4}-g_{-17}, g17g4g6g_{17}-g_{-4}-g_{-6}
Basis of centralizer intersected with cartan (dimension: 1): h6h3h2h_{6}-h_{3}-h_{2}
Cartan of centralizer (dimension: 1): h6h3h2h_{6}-h_{3}-h_{2}
Cartan-generating semisimple element: h6h3h2h_{6}-h_{3}-h_{2}
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & -2\\ \end{pmatrix}
Characteristic polynomial ad H: x34xx^3-4x
Factorization of characteristic polynomial of ad H: (x )(x -2)(x +2)
Eigenvalues of ad H: 00, 22, 2-2
3 eigenvectors of ad H: 1, 0, 0(1,0,0), 0, 1, 0(0,1,0), 0, 0, 1(0,0,1)
Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed: (190)\begin{pmatrix}1/90\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h6h3h2h_{6}-h_{3}-h_{2}
matching e: g6+g4g17g_{6}+g_{4}-g_{-17}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & -2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h6h3h2h_{6}-h_{3}-h_{2}
matching e: g6+g4g17g_{6}+g_{4}-g_{-17}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & -2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (360)\begin{pmatrix}360\\ \end{pmatrix}
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Unknown elements.
h=10h8+20h7+29h6+38h5+46h4+31h3+23h2+16h1e=x1g56+x7g46+x2g38+x4g37+x3g35+x6g34+x5g29e=x12g29+x13g34+x10g35+x11g37+x9g38+x14g46+x8g56\begin{array}{rcl}h&=&10h_{8}+20h_{7}+29h_{6}+38h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\\ e&=&x_{1} g_{56}+x_{7} g_{46}+x_{2} g_{38}+x_{4} g_{37}+x_{3} g_{35}+x_{6} g_{34}+x_{5} g_{29}\\ f&=&x_{12} g_{-29}+x_{13} g_{-34}+x_{10} g_{-35}+x_{11} g_{-37}+x_{9} g_{-38}+x_{14} g_{-46}+x_{8} g_{-56}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x8+x5x1210)h8+(x1x8+x3x10+x5x12+x6x1320)h7+(x1x8+x2x9+x3x10+x5x12+x6x13+x7x1429)h6+(x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1438)h5+(2x1x8+x2x9+x3x10+2x4x11+x6x13+2x7x1446)h4+(x1x8+x2x9+x3x10+x4x11+x7x1431)h3+(x1x8+x2x9+x4x11+x6x13+x7x1423)h2+(x2x9+x4x1116)h1[e,f] - h = \left(x_{1} x_{8} +x_{5} x_{12} -10\right)h_{8}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -20\right)h_{7}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -29\right)h_{6}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -38\right)h_{5}+\left(2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{6} x_{13} +2x_{7} x_{14} -46\right)h_{4}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{7} x_{14} -31\right)h_{3}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -23\right)h_{2}+\left(x_{2} x_{9} +x_{4} x_{11} -16\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x2x9+x4x11+x6x13+x7x1423=0x1x8+x2x9+x3x10+x4x11+x7x1431=02x1x8+x2x9+x3x10+2x4x11+x6x13+2x7x1446=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1438=0x1x8+x2x9+x3x10+x5x12+x6x13+x7x1429=0x1x8+x3x10+x5x12+x6x1320=0x1x8+x5x1210=0x2x9+x4x1116=0\begin{array}{rcl}x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -23&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{7} x_{14} -31&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{6} x_{13} +2x_{7} x_{14} -46&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -38&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -29&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -20&=&0\\x_{1} x_{8} +x_{5} x_{12} -10&=&0\\x_{2} x_{9} +x_{4} x_{11} -16&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+20h7+29h6+38h5+46h4+31h3+23h2+16h1e=x1g56+x7g46+x2g38+x4g37+x3g35+x6g34+x5g29f=g29+g34+g35+g37+g38+g46+g56\begin{array}{rcl}h&=&10h_{8}+20h_{7}+29h_{6}+38h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\\e&=&x_{1} g_{56}+x_{7} g_{46}+x_{2} g_{38}+x_{4} g_{37}+x_{3} g_{35}+x_{6} g_{34}+x_{5} g_{29}\\f&=&g_{-29}+g_{-34}+g_{-35}+g_{-37}+g_{-38}+g_{-46}+g_{-56}\end{array}
Matrix form of the system we are trying to solve: (11010111111001211201211111121110111101011010001000101000)[col. vect.]=(2331463829201016)\begin{pmatrix}1 & 1 & 0 & 1 & 0 & 1 & 1\\ 1 & 1 & 1 & 1 & 0 & 0 & 1\\ 2 & 1 & 1 & 2 & 0 & 1 & 2\\ 1 & 1 & 1 & 1 & 1 & 1 & 2\\ 1 & 1 & 1 & 0 & 1 & 1 & 1\\ 1 & 0 & 1 & 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 1 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}23\\ 31\\ 46\\ 38\\ 29\\ 20\\ 10\\ 16\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+20h7+29h6+38h5+46h4+31h3+23h2+16h1e=x1g56+x7g46+x2g38+x4g37+x3g35+x6g34+x5g29f=x12g29+x13g34+x10g35+x11g37+x9g38+x14g46+x8g56\begin{array}{rcl}h&=&10h_{8}+20h_{7}+29h_{6}+38h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\\ e&=&x_{1} g_{56}+x_{7} g_{46}+x_{2} g_{38}+x_{4} g_{37}+x_{3} g_{35}+x_{6} g_{34}+x_{5} g_{29}\\ f&=&x_{12} g_{-29}+x_{13} g_{-34}+x_{10} g_{-35}+x_{11} g_{-37}+x_{9} g_{-38}+x_{14} g_{-46}+x_{8} g_{-56}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x2x9+x4x11+x6x13+x7x1423=0x1x8+x2x9+x3x10+x4x11+x7x1431=02x1x8+x2x9+x3x10+2x4x11+x6x13+2x7x1446=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1438=0x1x8+x2x9+x3x10+x5x12+x6x13+x7x1429=0x1x8+x3x10+x5x12+x6x1320=0x1x8+x5x1210=0x2x9+x4x1116=0\begin{array}{rcl}x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -23&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{7} x_{14} -31&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{6} x_{13} +2x_{7} x_{14} -46&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -38&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -29&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} -20&=&0\\x_{1} x_{8} +x_{5} x_{12} -10&=&0\\x_{2} x_{9} +x_{4} x_{11} -16&=&0\\\end{array}

A136A^{36}_1
h-characteristic: (2, 0, 0, 0, 0, 0, 2, 0)
Length of the weight dual to h: 72
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A5+A1A^{1}_5+A^{1}_1 Containing regular semisimple subalgebra number 2: E6E^{1}_6
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 2V10ψ+8V8ψ+8V6ψ+15V4ψ+3V2ψ+14V02V_{10\psi}+8V_{8\psi}+8V_{6\psi}+15V_{4\psi}+3V_{2\psi}+14V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+20h7+28h6+36h5+44h4+30h3+22h2+16h1e=5g68+8g44+g35+9g34+8g32+5g15f=g15+g32+g34+g35+g44+g68\begin{array}{rcl}h&=&10h_{8}+20h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\\ e&=&5g_{68}+8g_{44}+g_{35}+9g_{34}+8g_{32}+5g_{15}\\ f&=&g_{-15}+g_{-32}+g_{-34}+g_{-35}+g_{-44}+g_{-68}\end{array}
Lie brackets of the above elements.
h=10h8+20h7+28h6+36h5+44h4+30h3+22h2+16h1e=5g68+8g44+g35+9g34+8g32+5g15f=g15+g32+g34+g35+g44+g68\begin{array}{rcl}[e, f]&=&10h_{8}+20h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\\ [h, e]&=&10g_{68}+16g_{44}+2g_{35}+18g_{34}+16g_{32}+10g_{15}\\ [h, f]&=&-2g_{-15}-2g_{-32}-2g_{-34}-2g_{-35}-2g_{-44}-2g_{-68}\end{array}
Centralizer type: G2G_2
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 14): g12g_{-12}, g5g_{-5}, g4g_{-4}, h4h_{4}, h5h_{5}, g4g_{4}, g5g_{5}, g12g_{12}, g17g6+g46g_{17}-g_{6}+g_{-46}, g25+g13+g39g_{25}+g_{13}+g_{-39}, g31+g20g33g_{31}+g_{20}-g_{-33}, g33g20g31g_{33}-g_{-20}-g_{-31}, g39+g13+g25g_{39}+g_{-13}+g_{-25}, g46g6+g17g_{46}-g_{-6}+g_{-17}
Basis of centralizer intersected with cartan (dimension: 2): h4-h_{4}, h5-h_{5}
Cartan of centralizer (dimension: 2): h4-h_{4}, h5-h_{5}
Cartan-generating semisimple element: 11h5h4-11h_{5}-h_{4}
adjoint action: (120000000000000021000000000000009000000000000000000000000000000000000000000009000000000000002100000000000000120000000000000011000000000000001000000000000000100000000000000100000000000000100000000000000011)\begin{pmatrix}12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 21 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -21 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -12 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -11\\ \end{pmatrix}
Characteristic polynomial ad H: x14888x12+271062x1037979068x8+2516250897x664718812404x4+62240270400x2x^{14}-888x^{12}+271062x^{10}-37979068x^8+2516250897x^6-64718812404x^4+62240270400x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -21)(x -12)(x -11)(x -10)(x -9)(x -1)(x +1)(x +9)(x +10)(x +11)(x +12)(x +21)
Eigenvalues of ad H: 00, 2121, 1212, 1111, 1010, 99, 11, 1-1, 9-9, 10-10, 11-11, 12-12, 21-21
14 eigenvectors of ad H: 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0,0,0,0,0), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,0,0,0,0,1,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,1,0,0,0), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,0,0,0,0,1), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,1,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,1,0,0,0,0,0,0,0)
Centralizer type: G^{1}_2
Reductive components (1 total):
Scalar product computed: (130160160190)\begin{pmatrix}1/30 & -1/60\\ -1/60 & 1/90\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
h4h_{4}
matching e: g4g_{4}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (1000000000000001000000000000002000000000000000000000000000000000000000000002000000000000001000000000000001000000000000000000000000000001000000000000001000000000000001000000000000001000000000000000)\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h52h4-h_{5}-2h_{4}
matching e: g33g20g31g_{33}-g_{-20}-g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (3000000000000000000000000000003000000000000000000000000000000000000000000003000000000000000000000000000003000000000000001000000000000001000000000000002000000000000002000000000000001000000000000001)\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h4h_{4}
matching e: g4g_{4}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (1000000000000001000000000000002000000000000000000000000000000000000000000002000000000000001000000000000001000000000000000000000000000001000000000000001000000000000001000000000000001000000000000000)\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h52h4-h_{5}-2h_{4}
matching e: g33g20g31g_{33}-g_{-20}-g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (3000000000000000000000000000003000000000000000000000000000000000000000000003000000000000000000000000000003000000000000001000000000000001000000000000002000000000000002000000000000001000000000000001)\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Elements in Cartan dual to root system: (2, 1), (-2, -1), (1, 1), (-1, -1), (3, 2), (-3, -2), (3, 1), (-3, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120180180360)\begin{pmatrix}120 & -180\\ -180 & 360\\ \end{pmatrix}
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Unknown elements.
h=10h8+20h7+28h6+36h5+44h4+30h3+22h2+16h1e=x1g68+x2g44+x6g35+x3g34+x4g32+x5g15e=x11g15+x10g32+x9g34+x12g35+x8g44+x7g68\begin{array}{rcl}h&=&10h_{8}+20h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\\ e&=&x_{1} g_{68}+x_{2} g_{44}+x_{6} g_{35}+x_{3} g_{34}+x_{4} g_{32}+x_{5} g_{15}\\ f&=&x_{11} g_{-15}+x_{10} g_{-32}+x_{9} g_{-34}+x_{12} g_{-35}+x_{8} g_{-44}+x_{7} g_{-68}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x7+x5x1110)h8+(x1x7+x3x9+x5x11+x6x1220)h7+(2x1x7+x3x9+x4x10+x6x1228)h6+(2x1x7+x2x8+x3x9+x4x10+x6x1236)h5+(2x1x7+2x2x8+x3x9+x4x10+x6x1244)h4+(x1x7+2x2x8+x4x10+x6x1230)h3+(x1x7+x2x8+x3x922)h2+(x2x8+x4x1016)h1[e,f] - h = \left(x_{1} x_{7} +x_{5} x_{11} -10\right)h_{8}+\left(x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -20\right)h_{7}+\left(2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -28\right)h_{6}+\left(2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -36\right)h_{5}+\left(2x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -44\right)h_{4}+\left(x_{1} x_{7} +2x_{2} x_{8} +x_{4} x_{10} +x_{6} x_{12} -30\right)h_{3}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} -22\right)h_{2}+\left(x_{2} x_{8} +x_{4} x_{10} -16\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x7+x2x8+x3x922=0x1x7+2x2x8+x4x10+x6x1230=02x1x7+2x2x8+x3x9+x4x10+x6x1244=02x1x7+x2x8+x3x9+x4x10+x6x1236=02x1x7+x3x9+x4x10+x6x1228=0x1x7+x3x9+x5x11+x6x1220=0x1x7+x5x1110=0x2x8+x4x1016=0\begin{array}{rcl}x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} -22&=&0\\x_{1} x_{7} +2x_{2} x_{8} +x_{4} x_{10} +x_{6} x_{12} -30&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -44&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -36&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -28&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -20&=&0\\x_{1} x_{7} +x_{5} x_{11} -10&=&0\\x_{2} x_{8} +x_{4} x_{10} -16&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+20h7+28h6+36h5+44h4+30h3+22h2+16h1e=x1g68+x2g44+x6g35+x3g34+x4g32+x5g15f=g15+g32+g34+g35+g44+g68\begin{array}{rcl}h&=&10h_{8}+20h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\\e&=&x_{1} g_{68}+x_{2} g_{44}+x_{6} g_{35}+x_{3} g_{34}+x_{4} g_{32}+x_{5} g_{15}\\f&=&g_{-15}+g_{-32}+g_{-34}+g_{-35}+g_{-44}+g_{-68}\end{array}
Matrix form of the system we are trying to solve: (111000120101221101211101201101101011100010010100)[col. vect.]=(2230443628201016)\begin{pmatrix}1 & 1 & 1 & 0 & 0 & 0\\ 1 & 2 & 0 & 1 & 0 & 1\\ 2 & 2 & 1 & 1 & 0 & 1\\ 2 & 1 & 1 & 1 & 0 & 1\\ 2 & 0 & 1 & 1 & 0 & 1\\ 1 & 0 & 1 & 0 & 1 & 1\\ 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}22\\ 30\\ 44\\ 36\\ 28\\ 20\\ 10\\ 16\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+20h7+28h6+36h5+44h4+30h3+22h2+16h1e=x1g68+x2g44+x6g35+x3g34+x4g32+x5g15f=x11g15+x10g32+x9g34+x12g35+x8g44+x7g68\begin{array}{rcl}h&=&10h_{8}+20h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\\ e&=&x_{1} g_{68}+x_{2} g_{44}+x_{6} g_{35}+x_{3} g_{34}+x_{4} g_{32}+x_{5} g_{15}\\ f&=&x_{11} g_{-15}+x_{10} g_{-32}+x_{9} g_{-34}+x_{12} g_{-35}+x_{8} g_{-44}+x_{7} g_{-68}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x7+x2x8+x3x922=0x1x7+2x2x8+x4x10+x6x1230=02x1x7+2x2x8+x3x9+x4x10+x6x1244=02x1x7+x2x8+x3x9+x4x10+x6x1236=02x1x7+x3x9+x4x10+x6x1228=0x1x7+x3x9+x5x11+x6x1220=0x1x7+x5x1110=0x2x8+x4x1016=0\begin{array}{rcl}x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} -22&=&0\\x_{1} x_{7} +2x_{2} x_{8} +x_{4} x_{10} +x_{6} x_{12} -30&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -44&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -36&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -28&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -20&=&0\\x_{1} x_{7} +x_{5} x_{11} -10&=&0\\x_{2} x_{8} +x_{4} x_{10} -16&=&0\\\end{array}

A136A^{36}_1
h-characteristic: (1, 0, 0, 1, 0, 0, 0, 1)
Length of the weight dual to h: 72
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A5+A1A^{1}_5+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V10ψ+4V9ψ+3V8ψ+2V7ψ+5V6ψ+8V5ψ+7V4ψ+4V3ψ+2V2ψ+4Vψ+6V0V_{10\psi}+4V_{9\psi}+3V_{8\psi}+2V_{7\psi}+5V_{6\psi}+8V_{5\psi}+7V_{4\psi}+4V_{3\psi}+2V_{2\psi}+4V_{\psi}+6V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+19h7+28h6+37h5+46h4+31h3+23h2+16h1e=5g50+g48+9g46+8g40+5g36+8g23f=g23+g36+g40+g46+g48+g50\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+37h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\\ e&=&5g_{50}+g_{48}+9g_{46}+8g_{40}+5g_{36}+8g_{23}\\ f&=&g_{-23}+g_{-36}+g_{-40}+g_{-46}+g_{-48}+g_{-50}\end{array}
Lie brackets of the above elements.
h=10h8+19h7+28h6+37h5+46h4+31h3+23h2+16h1e=5g50+g48+9g46+8g40+5g36+8g23f=g23+g36+g40+g46+g48+g50\begin{array}{rcl}[e, f]&=&10h_{8}+19h_{7}+28h_{6}+37h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\\ [h, e]&=&10g_{50}+2g_{48}+18g_{46}+16g_{40}+10g_{36}+16g_{23}\\ [h, f]&=&-2g_{-23}-2g_{-36}-2g_{-40}-2g_{-46}-2g_{-48}-2g_{-50}\end{array}
Centralizer type: A13+A1A^{3}_1+A_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 6): g6g_{-6}, h6h_{6}, h7+h5h3+h2h_{7}+h_{5}-h_{3}+h_{2}, g3g2g21g_{3}-g_{-2}-g_{-21}, g6g_{6}, g21+g2g3g_{21}+g_{2}-g_{-3}
Basis of centralizer intersected with cartan (dimension: 2): h7h5+h3h2-h_{7}-h_{5}+h_{3}-h_{2}, h6-h_{6}
Cartan of centralizer (dimension: 2): h7h5+h3h2-h_{7}-h_{5}+h_{3}-h_{2}, h6-h_{6}
Cartan-generating semisimple element: h711h6h5+h3h2-h_{7}-11h_{6}-h_{5}+h_{3}-h_{2}
adjoint action: (20000000000000000000002000000200000002)\begin{pmatrix}20 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & -20 & 0\\ 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
Characteristic polynomial ad H: x6404x4+1600x2x^6-404x^4+1600x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -20)(x -2)(x +2)(x +20)
Eigenvalues of ad H: 00, 2020, 22, 2-2, 20-20
6 eigenvectors of ad H: 0, 1, 0, 0, 0, 0(0,1,0,0,0,0), 0, 0, 1, 0, 0, 0(0,0,1,0,0,0), 1, 0, 0, 0, 0, 0(1,0,0,0,0,0), 0, 0, 0, 1, 0, 0(0,0,0,1,0,0), 0, 0, 0, 0, 0, 1(0,0,0,0,0,1), 0, 0, 0, 0, 1, 0(0,0,0,0,1,0)
Centralizer type: A^{3}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: (130)\begin{pmatrix}1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h6-h_{6}
matching e: g6g_{-6}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000000020000000)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h6-h_{6}
matching e: g6g_{-6}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000000020000000)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120)\begin{pmatrix}120\\ \end{pmatrix}

Scalar product computed: (190)\begin{pmatrix}1/90\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h7h6h5+h3h2-h_{7}-h_{6}-h_{5}+h_{3}-h_{2}
matching e: g3g2g21g_{3}-g_{-2}-g_{-21}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000200000000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h7h6h5+h3h2-h_{7}-h_{6}-h_{5}+h_{3}-h_{2}
matching e: g3g2g21g_{3}-g_{-2}-g_{-21}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000200000000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (360)\begin{pmatrix}360\\ \end{pmatrix}
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Unknown elements.
h=10h8+19h7+28h6+37h5+46h4+31h3+23h2+16h1e=x1g50+x6g48+x3g46+x2g40+x5g36+x4g23e=x10g23+x11g36+x8g40+x9g46+x12g48+x7g50\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+37h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\\ e&=&x_{1} g_{50}+x_{6} g_{48}+x_{3} g_{46}+x_{2} g_{40}+x_{5} g_{36}+x_{4} g_{23}\\ f&=&x_{10} g_{-23}+x_{11} g_{-36}+x_{8} g_{-40}+x_{9} g_{-46}+x_{12} g_{-48}+x_{7} g_{-50}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x7+x5x1110)h8+(x1x7+x2x8+x5x11+x6x1219)h7+(x1x7+x2x8+x3x9+x5x11+x6x1228)h6+(x1x7+x2x8+2x3x9+x5x11+x6x1237)h5+(x1x7+x2x8+2x3x9+x4x10+x5x11+2x6x1246)h4+(x1x7+x2x8+x3x9+x4x10+x6x1231)h3+(x1x7+x3x9+x4x10+x6x1223)h2+(x2x8+x4x1016)h1[e,f] - h = \left(x_{1} x_{7} +x_{5} x_{11} -10\right)h_{8}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{5} x_{11} +x_{6} x_{12} -19\right)h_{7}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -28\right)h_{6}+\left(x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -37\right)h_{5}+\left(x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +2x_{6} x_{12} -46\right)h_{4}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -31\right)h_{3}+\left(x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -23\right)h_{2}+\left(x_{2} x_{8} +x_{4} x_{10} -16\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x7+x3x9+x4x10+x6x1223=0x1x7+x2x8+x3x9+x4x10+x6x1231=0x1x7+x2x8+2x3x9+x4x10+x5x11+2x6x1246=0x1x7+x2x8+2x3x9+x5x11+x6x1237=0x1x7+x2x8+x3x9+x5x11+x6x1228=0x1x7+x2x8+x5x11+x6x1219=0x1x7+x5x1110=0x2x8+x4x1016=0\begin{array}{rcl}x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -23&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -31&=&0\\x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +2x_{6} x_{12} -46&=&0\\x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -37&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -28&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{5} x_{11} +x_{6} x_{12} -19&=&0\\x_{1} x_{7} +x_{5} x_{11} -10&=&0\\x_{2} x_{8} +x_{4} x_{10} -16&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+19h7+28h6+37h5+46h4+31h3+23h2+16h1e=x1g50+x6g48+x3g46+x2g40+x5g36+x4g23f=g23+g36+g40+g46+g48+g50\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+37h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\\e&=&x_{1} g_{50}+x_{6} g_{48}+x_{3} g_{46}+x_{2} g_{40}+x_{5} g_{36}+x_{4} g_{23}\\f&=&g_{-23}+g_{-36}+g_{-40}+g_{-46}+g_{-48}+g_{-50}\end{array}
Matrix form of the system we are trying to solve: (101101111101112112112011111011110011100010010100)[col. vect.]=(2331463728191016)\begin{pmatrix}1 & 0 & 1 & 1 & 0 & 1\\ 1 & 1 & 1 & 1 & 0 & 1\\ 1 & 1 & 2 & 1 & 1 & 2\\ 1 & 1 & 2 & 0 & 1 & 1\\ 1 & 1 & 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 0 & 1 & 1\\ 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}23\\ 31\\ 46\\ 37\\ 28\\ 19\\ 10\\ 16\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+19h7+28h6+37h5+46h4+31h3+23h2+16h1e=x1g50+x6g48+x3g46+x2g40+x5g36+x4g23f=x10g23+x11g36+x8g40+x9g46+x12g48+x7g50\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+37h_{5}+46h_{4}+31h_{3}+23h_{2}+16h_{1}\\ e&=&x_{1} g_{50}+x_{6} g_{48}+x_{3} g_{46}+x_{2} g_{40}+x_{5} g_{36}+x_{4} g_{23}\\ f&=&x_{10} g_{-23}+x_{11} g_{-36}+x_{8} g_{-40}+x_{9} g_{-46}+x_{12} g_{-48}+x_{7} g_{-50}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x7+x3x9+x4x10+x6x1223=0x1x7+x2x8+x3x9+x4x10+x6x1231=0x1x7+x2x8+2x3x9+x4x10+x5x11+2x6x1246=0x1x7+x2x8+2x3x9+x5x11+x6x1237=0x1x7+x2x8+x3x9+x5x11+x6x1228=0x1x7+x2x8+x5x11+x6x1219=0x1x7+x5x1110=0x2x8+x4x1016=0\begin{array}{rcl}x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -23&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -31&=&0\\x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +2x_{6} x_{12} -46&=&0\\x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -37&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -28&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{5} x_{11} +x_{6} x_{12} -19&=&0\\x_{1} x_{7} +x_{5} x_{11} -10&=&0\\x_{2} x_{8} +x_{4} x_{10} -16&=&0\\\end{array}

A135A^{35}_1
h-characteristic: (2, 0, 0, 0, 0, 1, 0, 1)
Length of the weight dual to h: 70
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A5A^{1}_5
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V10ψ+2V9ψ+7V8ψ+V6ψ+14V5ψ+7V4ψ+2V3ψ+V2ψ+17V0V_{10\psi}+2V_{9\psi}+7V_{8\psi}+V_{6\psi}+14V_{5\psi}+7V_{4\psi}+2V_{3\psi}+V_{2\psi}+17V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+19h7+28h6+36h5+44h4+30h3+22h2+16h1e=5g62+9g60+8g24+8g23+5g22f=g22+g23+g24+g60+g62\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\\ e&=&5g_{62}+9g_{60}+8g_{24}+8g_{23}+5g_{22}\\ f&=&g_{-22}+g_{-23}+g_{-24}+g_{-60}+g_{-62}\end{array}
Lie brackets of the above elements.
h=10h8+19h7+28h6+36h5+44h4+30h3+22h2+16h1e=5g62+9g60+8g24+8g23+5g22f=g22+g23+g24+g60+g62\begin{array}{rcl}[e, f]&=&10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\\ [h, e]&=&10g_{62}+18g_{60}+16g_{24}+16g_{23}+10g_{22}\\ [h, f]&=&-2g_{-22}-2g_{-23}-2g_{-24}-2g_{-60}-2g_{-62}\end{array}
Centralizer type: G2+A1G_2+A_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 17): g11g_{-11}, g7g_{-7}, g4g_{-4}, g3g_{-3}, h3h_{3}, h4h_{4}, h7h_{7}, g3g_{3}, g4g_{4}, g5+g2g31g_{5}+g_{2}-g_{-31}, g7g_{7}, g11g_{11}, g12g10+g25g_{12}-g_{10}+g_{-25}, g18+g17g19g_{18}+g_{-17}-g_{-19}, g19g17g18g_{19}-g_{17}-g_{-18}, g25g10+g12g_{25}-g_{-10}+g_{-12}, g31g2g5g_{31}-g_{-2}-g_{-5}
Basis of centralizer intersected with cartan (dimension: 3): h3-h_{3}, h7-h_{7}, h4-h_{4}
Cartan of centralizer (dimension: 3): h7-h_{7}, h4-h_{4}, h3-h_{3}
Cartan-generating semisimple element: 9h7+7h4h3-9h_{7}+7h_{4}-h_{3}
adjoint action: (60000000000000000018000000000000000001500000000000000000900000000000000000000000000000000000000000000000000000000000000000000000900000000000000000150000000000000000070000000000000000018000000000000000006000000000000000008000000000000000001000000000000000001000000000000000008000000000000000007)\begin{pmatrix}-6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -15 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 15 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7\\ \end{pmatrix}
Characteristic polynomial ad H: x17780x15+219222x1328262740x11+1824525153x957585871080x7+722428980624x5666639590400x3x^{17}-780x^{15}+219222x^{13}-28262740x^{11}+1824525153x^9-57585871080x^7+722428980624x^5-666639590400x^3
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -18)(x -15)(x -9)(x -8)(x -7)(x -6)(x -1)(x +1)(x +6)(x +7)(x +8)(x +9)(x +15)(x +18)
Eigenvalues of ad H: 00, 1818, 1515, 99, 88, 77, 66, 11, 1-1, 6-6, 7-7, 8-8, 9-9, 15-15, 18-18
17 eigenvectors of ad H: 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0), 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)
Centralizer type: G^{1}_2+A^{1}_1
Reductive components (2 total):
Scalar product computed: (130)\begin{pmatrix}1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h7-h_{7}
matching e: g7g_{-7}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h7-h_{7}
matching e: g7g_{-7}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120)\begin{pmatrix}120\\ \end{pmatrix}

Scalar product computed: (130160160190)\begin{pmatrix}1/30 & -1/60\\ -1/60 & 1/90\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
h4+h3h_{4}+h_{3}
matching e: g11g_{11}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2000000000000000000000000000000000001000000000000000001000000000000000000000000000000000000000000000000000000000000000000000001000000000000000001000000000000000001000000000000000000000000000000000002000000000000000000000000000000000001000000000000000001000000000000000000000000000000000001)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
h42h3-h_{4}-2h_{3}
matching e: g18+g17g19g_{18}+g_{-17}-g_{-19}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (3000000000000000000000000000000000000000000000000000003000000000000000000000000000000000000000000000000000000000000000000000003000000000000000000000000000000000001000000000000000000000000000000000003000000000000000001000000000000000002000000000000000002000000000000000001000000000000000001)\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h4+h3h_{4}+h_{3}
matching e: g11g_{11}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2000000000000000000000000000000000001000000000000000001000000000000000000000000000000000000000000000000000000000000000000000001000000000000000001000000000000000001000000000000000000000000000000000002000000000000000000000000000000000001000000000000000001000000000000000000000000000000000001)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
h42h3-h_{4}-2h_{3}
matching e: g18+g17g19g_{18}+g_{-17}-g_{-19}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (3000000000000000000000000000000000000000000000000000003000000000000000000000000000000000000000000000000000000000000000000000003000000000000000000000000000000000001000000000000000000000000000000000003000000000000000001000000000000000002000000000000000002000000000000000001000000000000000001)\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Elements in Cartan dual to root system: (2, 1), (-2, -1), (1, 1), (-1, -1), (3, 2), (-3, -2), (3, 1), (-3, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120180180360)\begin{pmatrix}120 & -180\\ -180 & 360\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=10h8+19h7+28h6+36h5+44h4+30h3+22h2+16h1e=x1g62+x3g60+x4g24+x2g23+x5g22e=x10g22+x7g23+x9g24+x8g60+x6g62\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\\ e&=&x_{1} g_{62}+x_{3} g_{60}+x_{4} g_{24}+x_{2} g_{23}+x_{5} g_{22}\\ f&=&x_{10} g_{-22}+x_{7} g_{-23}+x_{9} g_{-24}+x_{8} g_{-60}+x_{6} g_{-62}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x6+x5x1010)h8+(x1x6+x3x8+x5x1019)h7+(x1x6+2x3x8+x5x1028)h6+(2x1x6+2x3x8+x4x936)h5+(2x1x6+x2x7+2x3x8+x4x944)h4+(x1x6+x2x7+x3x8+x4x930)h3+(x1x6+x2x7+x3x822)h2+(x2x7+x4x916)h1[e,f] - h = \left(x_{1} x_{6} +x_{5} x_{10} -10\right)h_{8}+\left(x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -19\right)h_{7}+\left(x_{1} x_{6} +2x_{3} x_{8} +x_{5} x_{10} -28\right)h_{6}+\left(2x_{1} x_{6} +2x_{3} x_{8} +x_{4} x_{9} -36\right)h_{5}+\left(2x_{1} x_{6} +x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} -44\right)h_{4}+\left(x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} -30\right)h_{3}+\left(x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} -22\right)h_{2}+\left(x_{2} x_{7} +x_{4} x_{9} -16\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x6+x2x7+x3x822=0x1x6+x2x7+x3x8+x4x930=02x1x6+x2x7+2x3x8+x4x944=02x1x6+2x3x8+x4x936=0x1x6+2x3x8+x5x1028=0x1x6+x3x8+x5x1019=0x1x6+x5x1010=0x2x7+x4x916=0\begin{array}{rcl}x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} -22&=&0\\x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} -30&=&0\\2x_{1} x_{6} +x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} -44&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +x_{4} x_{9} -36&=&0\\x_{1} x_{6} +2x_{3} x_{8} +x_{5} x_{10} -28&=&0\\x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -19&=&0\\x_{1} x_{6} +x_{5} x_{10} -10&=&0\\x_{2} x_{7} +x_{4} x_{9} -16&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+19h7+28h6+36h5+44h4+30h3+22h2+16h1e=x1g62+x3g60+x4g24+x2g23+x5g22f=g22+g23+g24+g60+g62\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\\e&=&x_{1} g_{62}+x_{3} g_{60}+x_{4} g_{24}+x_{2} g_{23}+x_{5} g_{22}\\f&=&g_{-22}+g_{-23}+g_{-24}+g_{-60}+g_{-62}\end{array}
Matrix form of the system we are trying to solve: (1110011110212102021010201101011000101010)[col. vect.]=(2230443628191016)\begin{pmatrix}1 & 1 & 1 & 0 & 0\\ 1 & 1 & 1 & 1 & 0\\ 2 & 1 & 2 & 1 & 0\\ 2 & 0 & 2 & 1 & 0\\ 1 & 0 & 2 & 0 & 1\\ 1 & 0 & 1 & 0 & 1\\ 1 & 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}22\\ 30\\ 44\\ 36\\ 28\\ 19\\ 10\\ 16\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+19h7+28h6+36h5+44h4+30h3+22h2+16h1e=x1g62+x3g60+x4g24+x2g23+x5g22f=x10g22+x7g23+x9g24+x8g60+x6g62\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+16h_{1}\\ e&=&x_{1} g_{62}+x_{3} g_{60}+x_{4} g_{24}+x_{2} g_{23}+x_{5} g_{22}\\ f&=&x_{10} g_{-22}+x_{7} g_{-23}+x_{9} g_{-24}+x_{8} g_{-60}+x_{6} g_{-62}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x6+x2x7+x3x822=0x1x6+x2x7+x3x8+x4x930=02x1x6+x2x7+2x3x8+x4x944=02x1x6+2x3x8+x4x936=0x1x6+2x3x8+x5x1028=0x1x6+x3x8+x5x1019=0x1x6+x5x1010=0x2x7+x4x916=0\begin{array}{rcl}x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} -22&=&0\\x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} -30&=&0\\2x_{1} x_{6} +x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} -44&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +x_{4} x_{9} -36&=&0\\x_{1} x_{6} +2x_{3} x_{8} +x_{5} x_{10} -28&=&0\\x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -19&=&0\\x_{1} x_{6} +x_{5} x_{10} -10&=&0\\x_{2} x_{7} +x_{4} x_{9} -16&=&0\\\end{array}

A134A^{34}_1
h-characteristic: (0, 0, 1, 0, 0, 1, 0, 1)
Length of the weight dual to h: 68
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: D5+A2D^{1}_5+A^{1}_2
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V10ψ+2V9ψ+3V8ψ+6V7ψ+4V6ψ+6V5ψ+6V4ψ+4V3ψ+7V2ψ+4Vψ+3V0V_{10\psi}+2V_{9\psi}+3V_{8\psi}+6V_{7\psi}+4V_{6\psi}+6V_{5\psi}+6V_{4\psi}+4V_{3\psi}+7V_{2\psi}+4V_{\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+19h7+28h6+36h5+44h4+30h3+22h2+15h1e=23g5223g48+143g47+6g44+407g42+712g40+5514g368821g332921g27f=g272g33+4g364g40g42+g44+2g473g48+3g52\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+15h_{1}\\ e&=&2/3g_{52}-2/3g_{48}+14/3g_{47}+6g_{44}+40/7g_{42}+7/12g_{40}+55/14g_{36}-88/21g_{33}-29/21g_{27}\\ f&=&g_{-27}-2g_{-33}+4g_{-36}-4g_{-40}-g_{-42}+g_{-44}+2g_{-47}-3g_{-48}+3g_{-52}\end{array}
Lie brackets of the above elements.
h=10h8+19h7+28h6+36h5+44h4+30h3+22h2+15h1e=23g5223g48+143g47+6g44+407g42+712g40+5514g368821g332921g27f=g272g33+4g364g40g42+g44+2g473g48+3g52\begin{array}{rcl}[e, f]&=&10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+15h_{1}\\ [h, e]&=&4/3g_{52}-4/3g_{48}+28/3g_{47}+12g_{44}+80/7g_{42}+7/6g_{40}+55/7g_{36}-176/21g_{33}-58/21g_{27}\\ [h, f]&=&-2g_{-27}+4g_{-33}-8g_{-36}+8g_{-40}+2g_{-42}-2g_{-44}-4g_{-47}+6g_{-48}-6g_{-52}\end{array}
Centralizer type: A16A^{6}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 3): h7+2h5h1h_{7}+2h_{5}-h_{1}, g10+14g4+78g112g72g12+118g18g_{10}+1/4g_{4}+7/8g_{1}-1/2g_{-7}-2g_{-12}+11/8g_{-18}, g18+14g1278g7+12g12g4+118g10g_{18}+1/4g_{12}-7/8g_{7}+1/2g_{-1}-2g_{-4}+11/8g_{-10}
Basis of centralizer intersected with cartan (dimension: 1): h7+2h5h1h_{7}+2h_{5}-h_{1}
Cartan of centralizer (dimension: 1): h7+2h5h1h_{7}+2h_{5}-h_{1}
Cartan-generating semisimple element: h7+2h5h1h_{7}+2h_{5}-h_{1}
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Characteristic polynomial ad H: x34xx^3-4x
Factorization of characteristic polynomial of ad H: (x )(x -2)(x +2)
Eigenvalues of ad H: 00, 22, 2-2
3 eigenvectors of ad H: 1, 0, 0(1,0,0), 0, 0, 1(0,0,1), 0, 1, 0(0,1,0)
Centralizer type: A^{6}_1
Reductive components (1 total):
Scalar product computed: (1180)\begin{pmatrix}1/180\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h7+2h5h1h_{7}+2h_{5}-h_{1}
matching e: g18+14g1278g7+12g12g4+118g10g_{18}+1/4g_{12}-7/8g_{7}+1/2g_{-1}-2g_{-4}+11/8g_{-10}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h7+2h5h1h_{7}+2h_{5}-h_{1}
matching e: g18+14g1278g7+12g12g4+118g10g_{18}+1/4g_{12}-7/8g_{7}+1/2g_{-1}-2g_{-4}+11/8g_{-10}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (720)\begin{pmatrix}720\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=10h8+19h7+28h6+36h5+44h4+30h3+22h2+15h1e=x5g52+x6g48+x3g47+x1g44+x2g42+x8g40+x7g36+x4g33+x9g27e=x18g27+x13g33+x16g36+x17g40+x11g42+x10g44+x12g47+x15g48+x14g52\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+15h_{1}\\ e&=&x_{5} g_{52}+x_{6} g_{48}+x_{3} g_{47}+x_{1} g_{44}+x_{2} g_{42}+x_{8} g_{40}+x_{7} g_{36}+x_{4} g_{33}+x_{9} g_{27}\\ f&=&x_{18} g_{-27}+x_{13} g_{-33}+x_{16} g_{-36}+x_{17} g_{-40}+x_{11} g_{-42}+x_{10} g_{-44}+x_{12} g_{-47}+x_{15} g_{-48}+x_{14} g_{-52}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x16+x3x17+x4x18)g2+(x2x11+x7x1610)h8+(x2x11+x3x12+x6x15+x7x16+x8x1719)h7+(x2x11+x3x12+x4x13+x5x14+x6x15+x7x16+x8x17+x9x1828)h6+(x1x10+x2x11+x3x12+x4x13+2x5x14+x6x15+x7x16+x8x17+x9x1836)h5+(2x1x10+x2x11+x3x12+x4x13+2x5x14+2x6x15+x7x16+x8x17+x9x1844)h4+(2x1x10+x3x12+x4x13+x5x14+x6x15+x8x17+x9x1830)h3+(x1x10+x2x11+x3x12+x4x13+x5x14+x6x1522)h2+(x1x10+x3x12+x5x14+x8x1715)h1+(x7x11+x8x12+x9x13)g2[e,f] - h = \left(x_{2} x_{16} +x_{3} x_{17} +x_{4} x_{18} \right)g_{2}+\left(x_{2} x_{11} +x_{7} x_{16} -10\right)h_{8}+\left(x_{2} x_{11} +x_{3} x_{12} +x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} -19\right)h_{7}+\left(x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +x_{5} x_{14} +x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} +x_{9} x_{18} -28\right)h_{6}+\left(x_{1} x_{10} +x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +2x_{5} x_{14} +x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} +x_{9} x_{18} -36\right)h_{5}+\left(2x_{1} x_{10} +x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +2x_{5} x_{14} +2x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} +x_{9} x_{18} -44\right)h_{4}+\left(2x_{1} x_{10} +x_{3} x_{12} +x_{4} x_{13} +x_{5} x_{14} +x_{6} x_{15} +x_{8} x_{17} +x_{9} x_{18} -30\right)h_{3}+\left(x_{1} x_{10} +x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +x_{5} x_{14} +x_{6} x_{15} -22\right)h_{2}+\left(x_{1} x_{10} +x_{3} x_{12} +x_{5} x_{14} +x_{8} x_{17} -15\right)h_{1}+\left(x_{7} x_{11} +x_{8} x_{12} +x_{9} x_{13} \right)g_{-2}
The polynomial system that corresponds to finding the h, e, f triple:
x1x10+x3x12+x5x14+x8x1715=0x1x10+x2x11+x3x12+x4x13+x5x14+x6x1522=02x1x10+x3x12+x4x13+x5x14+x6x15+x8x17+x9x1830=02x1x10+x2x11+x3x12+x4x13+2x5x14+2x6x15+x7x16+x8x17+x9x1844=0x1x10+x2x11+x3x12+x4x13+2x5x14+x6x15+x7x16+x8x17+x9x1836=0x2x11+x3x12+x4x13+x5x14+x6x15+x7x16+x8x17+x9x1828=0x2x11+x3x12+x6x15+x7x16+x8x1719=0x2x11+x7x1610=0x2x16+x3x17+x4x18=0x7x11+x8x12+x9x13=0\begin{array}{rcl}x_{1} x_{10} +x_{3} x_{12} +x_{5} x_{14} +x_{8} x_{17} -15&=&0\\x_{1} x_{10} +x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +x_{5} x_{14} +x_{6} x_{15} -22&=&0\\2x_{1} x_{10} +x_{3} x_{12} +x_{4} x_{13} +x_{5} x_{14} +x_{6} x_{15} +x_{8} x_{17} +x_{9} x_{18} -30&=&0\\2x_{1} x_{10} +x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +2x_{5} x_{14} +2x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} +x_{9} x_{18} -44&=&0\\x_{1} x_{10} +x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +2x_{5} x_{14} +x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} +x_{9} x_{18} -36&=&0\\x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +x_{5} x_{14} +x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} +x_{9} x_{18} -28&=&0\\x_{2} x_{11} +x_{3} x_{12} +x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} -19&=&0\\x_{2} x_{11} +x_{7} x_{16} -10&=&0\\x_{2} x_{16} +x_{3} x_{17} +x_{4} x_{18} &=&0\\x_{7} x_{11} +x_{8} x_{12} +x_{9} x_{13} &=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+19h7+28h6+36h5+44h4+30h3+22h2+15h1e=x5g52+x6g48+x3g47+x1g44+x2g42+x8g40+x7g36+x4g33+x9g27f=g272g33+4g364g40g42+g44+2g473g48+3g52\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+15h_{1}\\e&=&x_{5} g_{52}+x_{6} g_{48}+x_{3} g_{47}+x_{1} g_{44}+x_{2} g_{42}+x_{8} g_{40}+x_{7} g_{36}+x_{4} g_{33}+x_{9} g_{27}\\f&=&g_{-27}-2g_{-33}+4g_{-36}-4g_{-40}-g_{-42}+g_{-44}+2g_{-47}-3g_{-48}+3g_{-52}\end{array}
Matrix form of the system we are trying to solve: (102030040112233000202233041212266441112263441012233441012003440010000400044100000000000122)[col. vect.]=(152230443628191000)\begin{pmatrix}1 & 0 & 2 & 0 & 3 & 0 & 0 & -4 & 0\\ 1 & -1 & 2 & -2 & 3 & -3 & 0 & 0 & 0\\ 2 & 0 & 2 & -2 & 3 & -3 & 0 & -4 & 1\\ 2 & -1 & 2 & -2 & 6 & -6 & 4 & -4 & 1\\ 1 & -1 & 2 & -2 & 6 & -3 & 4 & -4 & 1\\ 0 & -1 & 2 & -2 & 3 & -3 & 4 & -4 & 1\\ 0 & -1 & 2 & 0 & 0 & -3 & 4 & -4 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 4 & 0 & 0\\ 0 & 4 & -4 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -2\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}15\\ 22\\ 30\\ 44\\ 36\\ 28\\ 19\\ 10\\ 0\\ 0\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+19h7+28h6+36h5+44h4+30h3+22h2+15h1e=x5g52+x6g48+x3g47+x1g44+x2g42+x8g40+x7g36+x4g33+x9g27f=x18g27+x13g33+x16g36+x17g40+x11g42+x10g44+x12g47+x15g48+x14g52\begin{array}{rcl}h&=&10h_{8}+19h_{7}+28h_{6}+36h_{5}+44h_{4}+30h_{3}+22h_{2}+15h_{1}\\ e&=&x_{5} g_{52}+x_{6} g_{48}+x_{3} g_{47}+x_{1} g_{44}+x_{2} g_{42}+x_{8} g_{40}+x_{7} g_{36}+x_{4} g_{33}+x_{9} g_{27}\\ f&=&x_{18} g_{-27}+x_{13} g_{-33}+x_{16} g_{-36}+x_{17} g_{-40}+x_{11} g_{-42}+x_{10} g_{-44}+x_{12} g_{-47}+x_{15} g_{-48}+x_{14} g_{-52}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x10+x3x12+x5x14+x8x1715=0x1x10+x2x11+x3x12+x4x13+x5x14+x6x1522=02x1x10+x3x12+x4x13+x5x14+x6x15+x8x17+x9x1830=02x1x10+x2x11+x3x12+x4x13+2x5x14+2x6x15+x7x16+x8x17+x9x1844=0x1x10+x2x11+x3x12+x4x13+2x5x14+x6x15+x7x16+x8x17+x9x1836=0x2x11+x3x12+x4x13+x5x14+x6x15+x7x16+x8x17+x9x1828=0x2x11+x3x12+x6x15+x7x16+x8x1719=0x2x11+x7x1610=0x2x16+x3x17+x4x18=0x7x11+x8x12+x9x13=0\begin{array}{rcl}x_{1} x_{10} +x_{3} x_{12} +x_{5} x_{14} +x_{8} x_{17} -15&=&0\\x_{1} x_{10} +x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +x_{5} x_{14} +x_{6} x_{15} -22&=&0\\2x_{1} x_{10} +x_{3} x_{12} +x_{4} x_{13} +x_{5} x_{14} +x_{6} x_{15} +x_{8} x_{17} +x_{9} x_{18} -30&=&0\\2x_{1} x_{10} +x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +2x_{5} x_{14} +2x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} +x_{9} x_{18} -44&=&0\\x_{1} x_{10} +x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +2x_{5} x_{14} +x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} +x_{9} x_{18} -36&=&0\\x_{2} x_{11} +x_{3} x_{12} +x_{4} x_{13} +x_{5} x_{14} +x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} +x_{9} x_{18} -28&=&0\\x_{2} x_{11} +x_{3} x_{12} +x_{6} x_{15} +x_{7} x_{16} +x_{8} x_{17} -19&=&0\\x_{2} x_{11} +x_{7} x_{16} -10&=&0\\x_{2} x_{16} +x_{3} x_{17} +x_{4} x_{18} &=&0\\x_{7} x_{11} +x_{8} x_{12} +x_{9} x_{13} &=&0\\\end{array}

A132A^{32}_1
h-characteristic: (0, 2, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 64
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: D4+4A1D^{1}_4+4A^{1}_1 Containing regular semisimple subalgebra number 2: D4+A2D^{1}_4+A^{1}_2 Containing regular semisimple subalgebra number 3: D5+2A1D^{1}_5+2A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V10ψ+6V8ψ+14V6ψ+7V4ψ+14V2ψ+8V0V_{10\psi}+6V_{8\psi}+14V_{6\psi}+7V_{4\psi}+14V_{2\psi}+8V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+18h7+26h6+34h5+42h4+28h3+22h2+14h1e=6g81+g53+6g44+g41+g39+g38+10g22+6g18f=g18+g22+g38+g39+g41+g44+g53+g81\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+22h_{2}+14h_{1}\\ e&=&6g_{81}+g_{53}+6g_{44}+g_{41}+g_{39}+g_{38}+10g_{22}+6g_{18}\\ f&=&g_{-18}+g_{-22}+g_{-38}+g_{-39}+g_{-41}+g_{-44}+g_{-53}+g_{-81}\end{array}
Lie brackets of the above elements.
h=10h8+18h7+26h6+34h5+42h4+28h3+22h2+14h1e=6g81+g53+6g44+g41+g39+g38+10g22+6g18f=g18+g22+g38+g39+g41+g44+g53+g81\begin{array}{rcl}[e, f]&=&10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+22h_{2}+14h_{1}\\ [h, e]&=&12g_{81}+2g_{53}+12g_{44}+2g_{41}+2g_{39}+2g_{38}+20g_{22}+12g_{18}\\ [h, f]&=&-2g_{-18}-2g_{-22}-2g_{-38}-2g_{-39}-2g_{-41}-2g_{-44}-2g_{-53}-2g_{-81}\end{array}
Centralizer type: A22A^{2}_2
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 8): g4g1g1+g4g_{4}-g_{1}-g_{-1}+g_{-4}, g7+g1+g1+g7g_{7}+g_{1}+g_{-1}+g_{-7}, g11+g9g9g11g_{11}+g_{9}-g_{-9}-g_{-11}, g16+g3+g3+g16g_{16}+g_{3}+g_{-3}+g_{-16}, g21g20g20+g21g_{21}-g_{20}-g_{-20}+g_{-21}, g28g13+g13g28g_{28}-g_{13}+g_{-13}-g_{-28}, g35g32g32+g35g_{35}-g_{32}-g_{-32}+g_{-35}, g40g27+g27g40g_{40}-g_{27}+g_{-27}-g_{-40}
Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 2): 2g402g35+2g3212g28+2g2712g21+12g202g16+12g132g112g9g42g3+g1+g12g3g4+2g9+2g1112g132g16+12g2012g212g27+12g28+2g322g35+2g40-2g_{40}-2g_{35}+2g_{32}-1/2g_{28}+2g_{27}-1/2g_{21}+1/2g_{20}-2g_{16}+1/2g_{13}-2g_{11}-2g_{9}-g_{4}-2g_{3}+g_{1}+g_{-1}-2g_{-3}-g_{-4}+2g_{-9}+2g_{-11}-1/2g_{-13}-2g_{-16}+1/2g_{-20}-1/2g_{-21}-2g_{-27}+1/2g_{-28}+2g_{-32}-2g_{-35}+2g_{-40}, g40+g35g3212g28g2712g21+12g20+g16+12g13+g11+g9g7+g3g1g1+g3g7g9g1112g13+g16+12g2012g21+g27+12g28g32+g35g40g_{40}+g_{35}-g_{32}-1/2g_{28}-g_{27}-1/2g_{21}+1/2g_{20}+g_{16}+1/2g_{13}+g_{11}+g_{9}-g_{7}+g_{3}-g_{1}-g_{-1}+g_{-3}-g_{-7}-g_{-9}-g_{-11}-1/2g_{-13}+g_{-16}+1/2g_{-20}-1/2g_{-21}+g_{-27}+1/2g_{-28}-g_{-32}+g_{-35}-g_{-40}
Cartan-generating semisimple element: g40+g35g32+g28g27+g21g20+g16g13+g11+g9+g7+g4+g3+g3+g4+g7g9g11+g13+g16g20+g21+g27g28g32+g35g40g_{40}+g_{35}-g_{32}+g_{28}-g_{27}+g_{21}-g_{20}+g_{16}-g_{13}+g_{11}+g_{9}+g_{7}+g_{4}+g_{3}+g_{-3}+g_{-4}+g_{-7}-g_{-9}-g_{-11}+g_{-13}+g_{-16}-g_{-20}+g_{-21}+g_{-27}-g_{-28}-g_{-32}+g_{-35}-g_{-40}
adjoint action: (0022220000002222210111112110111111110211111120111211110112111110)\begin{pmatrix}0 & 0 & 2 & -2 & 2 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & -2 & 2 & -2\\ 2 & -1 & 0 & -1 & 1 & -1 & -1 & 1\\ 2 & -1 & -1 & 0 & 1 & -1 & 1 & -1\\ -1 & -1 & 1 & -1 & 0 & 2 & -1 & 1\\ -1 & -1 & 1 & -1 & 2 & 0 & 1 & -1\\ 1 & -2 & -1 & -1 & 1 & 1 & 0 & 1\\ 1 & -2 & -1 & -1 & 1 & 1 & 1 & 0\\ \end{pmatrix}
Characteristic polynomial ad H: x86x6+9x44x2x^8-6x^6+9x^4-4x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)
Eigenvalues of ad H: 00, 22, 11, 1-1, 2-2
8 eigenvectors of ad H: 2/3, 4/3, 0, 0, 1, 1, 0, 0(23,43,0,0,1,1,0,0), 1/3, -1/3, 1, 1, 0, 0, 1, 1(13,13,1,1,0,0,1,1), 0, 0, 0, 0, 1/2, 1/2, 1, 1(0,0,0,0,12,12,1,1), 2, 0, 5/2, 3/2, 1, 1, 0, 0(2,0,52,32,1,1,0,0), 0, 0, 0, 0, 0, 0, 1, 1(0,0,0,0,0,0,1,1), 2, 2, 1, 0, 0, 2, 1, 0(2,2,1,0,0,2,1,0), -2, -2, 0, 1, 0, -2, 0, 1(2,2,0,1,0,2,0,1), 1, 1, 0, 0, 0, 1, 0, 0(1,1,0,0,0,1,0,0)
Centralizer type: A^{2}_2
Reductive components (1 total):
Scalar product computed: (16011201120160)\begin{pmatrix}1/60 & -1/120\\ -1/120 & 1/60\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
2g40+2g352g32+12g282g27+12g2112g20+2g1612g13+2g11+2g9+g4+2g3g1g1+2g3+g42g92g11+12g13+2g1612g20+12g21+2g2712g282g32+2g352g402g_{40}+2g_{35}-2g_{32}+1/2g_{28}-2g_{27}+1/2g_{21}-1/2g_{20}+2g_{16}-1/2g_{13}+2g_{11}+2g_{9}+g_{4}+2g_{3}-g_{1}-g_{-1}+2g_{-3}+g_{-4}-2g_{-9}-2g_{-11}+1/2g_{-13}+2g_{-16}-1/2g_{-20}+1/2g_{-21}+2g_{-27}-1/2g_{-28}-2g_{-32}+2g_{-35}-2g_{-40}
matching e: g40+g35g32+12g28g27+12g2112g20+34g1612g13+54g11+54g9+g4+34g3g1g1+34g3+g454g954g11+12g13+34g1612g20+12g21+g2712g28g32+g35g40g_{40}+g_{35}-g_{32}+1/2g_{28}-g_{27}+1/2g_{21}-1/2g_{20}+3/4g_{16}-1/2g_{13}+5/4g_{11}+5/4g_{9}+g_{4}+3/4g_{3}-g_{1}-g_{-1}+3/4g_{-3}+g_{-4}-5/4g_{-9}-5/4g_{-11}+1/2g_{-13}+3/4g_{-16}-1/2g_{-20}+1/2g_{-21}+g_{-27}-1/2g_{-28}-g_{-32}+g_{-35}-g_{-40}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0044110000001144420222121242202212121212220122121222102224121222012412122210)\begin{pmatrix}0 & 0 & 4 & -4 & 1 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & -1 & 4 & -4\\ 4 & -2 & 0 & -2 & 2 & -2 & -1/2 & 1/2\\ 4 & -2 & -2 & 0 & 2 & -2 & 1/2 & -1/2\\ -1/2 & -1/2 & 2 & -2 & 0 & 1 & -2 & 2\\ -1/2 & -1/2 & 2 & -2 & 1 & 0 & 2 & -2\\ 2 & -4 & -1/2 & -1/2 & 2 & 2 & 0 & -1\\ 2 & -4 & -1/2 & -1/2 & 2 & 2 & -1 & 0\\ \end{pmatrix}
g40g35+g32+12g28+g27+12g2112g20g1612g13g11g9+g7g3+g1+g1g3+g7+g9+g11+12g13g1612g20+12g21g2712g28+g32g35+g40-g_{40}-g_{35}+g_{32}+1/2g_{28}+g_{27}+1/2g_{21}-1/2g_{20}-g_{16}-1/2g_{13}-g_{11}-g_{9}+g_{7}-g_{3}+g_{1}+g_{-1}-g_{-3}+g_{-7}+g_{-9}+g_{-11}+1/2g_{-13}-g_{-16}-1/2g_{-20}+1/2g_{-21}-g_{-27}-1/2g_{-28}+g_{-32}-g_{-35}+g_{-40}
matching e: g40+g35g32g27+g27g32+g35g40g_{40}+g_{35}-g_{32}-g_{27}+g_{-27}-g_{-32}+g_{-35}-g_{-40}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0022110000001122210111121221101112121212110111121211101112121211021212121120)\begin{pmatrix}0 & 0 & -2 & 2 & 1 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & -1 & -2 & 2\\ -2 & 1 & 0 & 1 & -1 & 1 & -1/2 & 1/2\\ -2 & 1 & 1 & 0 & -1 & 1 & 1/2 & -1/2\\ -1/2 & -1/2 & -1 & 1 & 0 & 1 & 1 & -1\\ -1/2 & -1/2 & -1 & 1 & 1 & 0 & -1 & 1\\ -1 & 2 & -1/2 & -1/2 & -1 & -1 & 0 & 2\\ -1 & 2 & -1/2 & -1/2 & -1 & -1 & 2 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
2g40+2g352g32+12g282g27+12g2112g20+2g1612g13+2g11+2g9+g4+2g3g1g1+2g3+g42g92g11+12g13+2g1612g20+12g21+2g2712g282g32+2g352g402g_{40}+2g_{35}-2g_{32}+1/2g_{28}-2g_{27}+1/2g_{21}-1/2g_{20}+2g_{16}-1/2g_{13}+2g_{11}+2g_{9}+g_{4}+2g_{3}-g_{1}-g_{-1}+2g_{-3}+g_{-4}-2g_{-9}-2g_{-11}+1/2g_{-13}+2g_{-16}-1/2g_{-20}+1/2g_{-21}+2g_{-27}-1/2g_{-28}-2g_{-32}+2g_{-35}-2g_{-40}
matching e: g40+g35g32+12g28g27+12g2112g20+34g1612g13+54g11+54g9+g4+34g3g1g1+34g3+g454g954g11+12g13+34g1612g20+12g21+g2712g28g32+g35g40g_{40}+g_{35}-g_{32}+1/2g_{28}-g_{27}+1/2g_{21}-1/2g_{20}+3/4g_{16}-1/2g_{13}+5/4g_{11}+5/4g_{9}+g_{4}+3/4g_{3}-g_{1}-g_{-1}+3/4g_{-3}+g_{-4}-5/4g_{-9}-5/4g_{-11}+1/2g_{-13}+3/4g_{-16}-1/2g_{-20}+1/2g_{-21}+g_{-27}-1/2g_{-28}-g_{-32}+g_{-35}-g_{-40}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0044110000001144420222121242202212121212220122121222102224121222012412122210)\begin{pmatrix}0 & 0 & 4 & -4 & 1 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & -1 & 4 & -4\\ 4 & -2 & 0 & -2 & 2 & -2 & -1/2 & 1/2\\ 4 & -2 & -2 & 0 & 2 & -2 & 1/2 & -1/2\\ -1/2 & -1/2 & 2 & -2 & 0 & 1 & -2 & 2\\ -1/2 & -1/2 & 2 & -2 & 1 & 0 & 2 & -2\\ 2 & -4 & -1/2 & -1/2 & 2 & 2 & 0 & -1\\ 2 & -4 & -1/2 & -1/2 & 2 & 2 & -1 & 0\\ \end{pmatrix}
g40g35+g32+12g28+g27+12g2112g20g1612g13g11g9+g7g3+g1+g1g3+g7+g9+g11+12g13g1612g20+12g21g2712g28+g32g35+g40-g_{40}-g_{35}+g_{32}+1/2g_{28}+g_{27}+1/2g_{21}-1/2g_{20}-g_{16}-1/2g_{13}-g_{11}-g_{9}+g_{7}-g_{3}+g_{1}+g_{-1}-g_{-3}+g_{-7}+g_{-9}+g_{-11}+1/2g_{-13}-g_{-16}-1/2g_{-20}+1/2g_{-21}-g_{-27}-1/2g_{-28}+g_{-32}-g_{-35}+g_{-40}
matching e: g40+g35g32g27+g27g32+g35g40g_{40}+g_{35}-g_{32}-g_{27}+g_{-27}-g_{-32}+g_{-35}-g_{-40}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0022110000001122210111121221101112121212110111121211101112121211021212121120)\begin{pmatrix}0 & 0 & -2 & 2 & 1 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & -1 & -2 & 2\\ -2 & 1 & 0 & 1 & -1 & 1 & -1/2 & 1/2\\ -2 & 1 & 1 & 0 & -1 & 1 & 1/2 & -1/2\\ -1/2 & -1/2 & -1 & 1 & 0 & 1 & 1 & -1\\ -1/2 & -1/2 & -1 & 1 & 1 & 0 & -1 & 1\\ -1 & 2 & -1/2 & -1/2 & -1 & -1 & 0 & 2\\ -1 & 2 & -1/2 & -1/2 & -1 & -1 & 2 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240120120240)\begin{pmatrix}240 & -120\\ -120 & 240\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=10h8+18h7+26h6+34h5+42h4+28h3+22h2+14h1e=x1g81+x5g53+x3g44+x8g41+x6g39+x7g38+x2g22+x4g18e=x12g18+x10g22+x15g38+x14g39+x16g41+x11g44+x13g53+x9g81\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+22h_{2}+14h_{1}\\ e&=&x_{1} g_{81}+x_{5} g_{53}+x_{3} g_{44}+x_{8} g_{41}+x_{6} g_{39}+x_{7} g_{38}+x_{2} g_{22}+x_{4} g_{18}\\ f&=&x_{12} g_{-18}+x_{10} g_{-22}+x_{15} g_{-38}+x_{14} g_{-39}+x_{16} g_{-41}+x_{11} g_{-44}+x_{13} g_{-53}+x_{9} g_{-81}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x1010)h8+(x1x9+x2x10+x5x13+x8x1618)h7+(2x1x9+x2x10+x5x13+x6x14+x7x15+x8x1626)h6+(3x1x9+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1634)h5+(3x1x9+2x3x11+x4x12+2x5x13+2x6x14+x7x15+x8x1642)h4+(2x1x9+2x3x11+x5x13+x6x14+x7x15+x8x1628)h3+(x1x9+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1622)h2+(x1x9+x3x11+x5x13+x7x1514)h1[e,f] - h = \left(x_{2} x_{10} -10\right)h_{8}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{8} x_{16} -18\right)h_{7}+\left(2x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -26\right)h_{6}+\left(3x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -34\right)h_{5}+\left(3x_{1} x_{9} +2x_{3} x_{11} +x_{4} x_{12} +2x_{5} x_{13} +2x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -42\right)h_{4}+\left(2x_{1} x_{9} +2x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -28\right)h_{3}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -22\right)h_{2}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} -14\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x9+x3x11+x5x13+x7x1514=0x1x9+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1622=02x1x9+2x3x11+x5x13+x6x14+x7x15+x8x1628=03x1x9+2x3x11+x4x12+2x5x13+2x6x14+x7x15+x8x1642=03x1x9+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1634=02x1x9+x2x10+x5x13+x6x14+x7x15+x8x1626=0x1x9+x2x10+x5x13+x8x1618=0x2x1010=0\begin{array}{rcl}x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} -14&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -22&=&0\\2x_{1} x_{9} +2x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -28&=&0\\3x_{1} x_{9} +2x_{3} x_{11} +x_{4} x_{12} +2x_{5} x_{13} +2x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -42&=&0\\3x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -34&=&0\\2x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -26&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{8} x_{16} -18&=&0\\x_{2} x_{10} -10&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+18h7+26h6+34h5+42h4+28h3+22h2+14h1e=x1g81+x5g53+x3g44+x8g41+x6g39+x7g38+x2g22+x4g18f=g18+g22+g38+g39+g41+g44+g53+g81\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+22h_{2}+14h_{1}\\e&=&x_{1} g_{81}+x_{5} g_{53}+x_{3} g_{44}+x_{8} g_{41}+x_{6} g_{39}+x_{7} g_{38}+x_{2} g_{22}+x_{4} g_{18}\\f&=&g_{-18}+g_{-22}+g_{-38}+g_{-39}+g_{-41}+g_{-44}+g_{-53}+g_{-81}\end{array}
Matrix form of the system we are trying to solve: (1010101010111111202011113021221130111111210011111100100101000000)[col. vect.]=(1422284234261810)\begin{pmatrix}1 & 0 & 1 & 0 & 1 & 0 & 1 & 0\\ 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1\\ 2 & 0 & 2 & 0 & 1 & 1 & 1 & 1\\ 3 & 0 & 2 & 1 & 2 & 2 & 1 & 1\\ 3 & 0 & 1 & 1 & 1 & 1 & 1 & 1\\ 2 & 1 & 0 & 0 & 1 & 1 & 1 & 1\\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 22\\ 28\\ 42\\ 34\\ 26\\ 18\\ 10\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+18h7+26h6+34h5+42h4+28h3+22h2+14h1e=x1g81+x5g53+x3g44+x8g41+x6g39+x7g38+x2g22+x4g18f=x12g18+x10g22+x15g38+x14g39+x16g41+x11g44+x13g53+x9g81\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+22h_{2}+14h_{1}\\ e&=&x_{1} g_{81}+x_{5} g_{53}+x_{3} g_{44}+x_{8} g_{41}+x_{6} g_{39}+x_{7} g_{38}+x_{2} g_{22}+x_{4} g_{18}\\ f&=&x_{12} g_{-18}+x_{10} g_{-22}+x_{15} g_{-38}+x_{14} g_{-39}+x_{16} g_{-41}+x_{11} g_{-44}+x_{13} g_{-53}+x_{9} g_{-81}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x9+x3x11+x5x13+x7x1514=0x1x9+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1622=02x1x9+2x3x11+x5x13+x6x14+x7x15+x8x1628=03x1x9+2x3x11+x4x12+2x5x13+2x6x14+x7x15+x8x1642=03x1x9+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1634=02x1x9+x2x10+x5x13+x6x14+x7x15+x8x1626=0x1x9+x2x10+x5x13+x8x1618=0x2x1010=0\begin{array}{rcl}x_{1} x_{9} +x_{3} x_{11} +x_{5} x_{13} +x_{7} x_{15} -14&=&0\\x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -22&=&0\\2x_{1} x_{9} +2x_{3} x_{11} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -28&=&0\\3x_{1} x_{9} +2x_{3} x_{11} +x_{4} x_{12} +2x_{5} x_{13} +2x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -42&=&0\\3x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -34&=&0\\2x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -26&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{5} x_{13} +x_{8} x_{16} -18&=&0\\x_{2} x_{10} -10&=&0\\\end{array}

A131A^{31}_1
h-characteristic: (0, 0, 0, 1, 0, 0, 0, 2)
Length of the weight dual to h: 62
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: D4+3A1D^{1}_4+3A^{1}_1 Containing regular semisimple subalgebra number 2: D5+A1D^{1}_5+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V10ψ+3V8ψ+6V7ψ+8V6ψ+6V5ψ+3V4ψ+2V3ψ+7V2ψ+10Vψ+6V0V_{10\psi}+3V_{8\psi}+6V_{7\psi}+8V_{6\psi}+6V_{5\psi}+3V_{4\psi}+2V_{3\psi}+7V_{2\psi}+10V_{\psi}+6V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+18h7+26h6+34h5+42h4+28h3+21h2+14h1e=6g71+g54+g53+6g52+g44+6g39+10g15f=g15+g39+g44+g52+g53+g54+g71\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\\ e&=&6g_{71}+g_{54}+g_{53}+6g_{52}+g_{44}+6g_{39}+10g_{15}\\ f&=&g_{-15}+g_{-39}+g_{-44}+g_{-52}+g_{-53}+g_{-54}+g_{-71}\end{array}
Lie brackets of the above elements.
h=10h8+18h7+26h6+34h5+42h4+28h3+21h2+14h1e=6g71+g54+g53+6g52+g44+6g39+10g15f=g15+g39+g44+g52+g53+g54+g71\begin{array}{rcl}[e, f]&=&10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\\ [h, e]&=&12g_{71}+2g_{54}+2g_{53}+12g_{52}+2g_{44}+12g_{39}+20g_{15}\\ [h, f]&=&-2g_{-15}-2g_{-39}-2g_{-44}-2g_{-52}-2g_{-53}-2g_{-54}-2g_{-71}\end{array}
Centralizer type: A18+A1A^{8}_1+A_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 6): g2g_{-2}, h2h_{2}, g2g_{2}, g5g1g1+g5g_{5}-g_{1}-g_{-1}+g_{-5}, g14+g3+g3+g14g_{14}+g_{3}+g_{-3}+g_{-14}, g21g9+g9g21g_{21}-g_{9}+g_{-9}-g_{-21}
Basis of centralizer intersected with cartan (dimension: 1): h2-h_{2}
Cartan of centralizer (dimension: 2): g2h2g2-g_{2}-h_{2}-g_{-2}, g21g14+g9g5g3+g1+g1g3g5g9g14+g21-g_{21}-g_{14}+g_{9}-g_{5}-g_{3}+g_{1}+g_{-1}-g_{-3}-g_{-5}-g_{-9}-g_{-14}+g_{-21}
Cartan-generating semisimple element: g21+g14g9+g5+g3+g2g1+h2g1+g2+g3+g5+g9+g14g21g_{21}+g_{14}-g_{9}+g_{5}+g_{3}+g_{2}-g_{1}+h_{2}-g_{-1}+g_{-2}+g_{-3}+g_{-5}+g_{-9}+g_{-14}-g_{-21}
adjoint action: (220000101000022000000011000101000110)\begin{pmatrix}-2 & 2 & 0 & 0 & 0 & 0\\ 1 & 0 & -1 & 0 & 0 & 0\\ 0 & -2 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & -1\\ 0 & 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 0 & -1 & 1 & 0\\ \end{pmatrix}
Characteristic polynomial ad H: x69x4+8x2x^6-9x^4+8x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -1)(x +1)(x^2-8)
Eigenvalues of ad H: 00, 11, 1-1, 222\sqrt{2}, 22-2\sqrt{2}
6 eigenvectors of ad H: 1, 1, 1, 0, 0, 0(1,1,1,0,0,0), 0, 0, 0, 1, 1, 1(0,0,0,1,1,1), 0, 0, 0, 0, 1, 1(0,0,0,0,1,1), 0, 0, 0, 1, 0, 1(0,0,0,1,0,1), 2\sqrt{2}-3, -\sqrt{2}+1, 1, 0, 0, 0(223,2+1,1,0,0,0), -2\sqrt{2}-3, \sqrt{2}+1, 1, 0, 0, 0(223,2+1,1,0,0,0)
Centralizer type: A^{8}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: (1240)\begin{pmatrix}1/240\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
2g21+2g142g9+2g5+2g32g12g1+2g3+2g5+2g9+2g142g212g_{21}+2g_{14}-2g_{9}+2g_{5}+2g_{3}-2g_{1}-2g_{-1}+2g_{-3}+2g_{-5}+2g_{-9}+2g_{-14}-2g_{-21}
matching e: g21+g14g9+g3+g3+g9+g14g21g_{21}+g_{14}-g_{9}+g_{3}+g_{-3}+g_{-9}+g_{-14}-g_{-21}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000022000202000220)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & -2\\ 0 & 0 & 0 & -2 & 0 & 2\\ 0 & 0 & 0 & -2 & 2 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
2g21+2g142g9+2g5+2g32g12g1+2g3+2g5+2g9+2g142g212g_{21}+2g_{14}-2g_{9}+2g_{5}+2g_{3}-2g_{1}-2g_{-1}+2g_{-3}+2g_{-5}+2g_{-9}+2g_{-14}-2g_{-21}
matching e: g21+g14g9+g3+g3+g9+g14g21g_{21}+g_{14}-g_{9}+g_{3}+g_{-3}+g_{-9}+g_{-14}-g_{-21}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000022000202000220)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & -2\\ 0 & 0 & 0 & -2 & 0 & 2\\ 0 & 0 & 0 & -2 & 2 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (960)\begin{pmatrix}960\\ \end{pmatrix}

Scalar product computed: (130)\begin{pmatrix}1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
122g2+122h2+122g21/2\sqrt{2}g_{2}+1/2\sqrt{2}h_{2}+1/2\sqrt{2}g_{-2}
matching e: g2+(2+1)h2+(223)g2g_{2}+\left(-\sqrt{2}+1\right)h_{2}+\left(2\sqrt{2}-3\right)g_{-2}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2200001220122000022000000000000000000000)\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & -1/2\sqrt{2} & 0 & 0 & 0\\ 0 & -\sqrt{2} & \sqrt{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
122g2+122h2+122g21/2\sqrt{2}g_{2}+1/2\sqrt{2}h_{2}+1/2\sqrt{2}g_{-2}
matching e: g2+(2+1)h2+(223)g2g_{2}+\left(-\sqrt{2}+1\right)h_{2}+\left(2\sqrt{2}-3\right)g_{-2}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2200001220122000022000000000000000000000)\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & -1/2\sqrt{2} & 0 & 0 & 0\\ 0 & -\sqrt{2} & \sqrt{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120)\begin{pmatrix}120\\ \end{pmatrix}
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Unknown elements.
h=10h8+18h7+26h6+34h5+42h4+28h3+21h2+14h1e=x1g71+x6g54+x7g53+x3g52+x5g44+x4g39+x2g15e=x9g15+x11g39+x12g44+x10g52+x14g53+x13g54+x8g71\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\\ e&=&x_{1} g_{71}+x_{6} g_{54}+x_{7} g_{53}+x_{3} g_{52}+x_{5} g_{44}+x_{4} g_{39}+x_{2} g_{15}\\ f&=&x_{9} g_{-15}+x_{11} g_{-39}+x_{12} g_{-44}+x_{10} g_{-52}+x_{14} g_{-53}+x_{13} g_{-54}+x_{8} g_{-71}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x910)h8+(x1x8+x2x9+x6x13+x7x1418)h7+(2x1x8+x3x10+x4x11+x6x13+x7x1426)h6+(2x1x8+2x3x10+x4x11+x5x12+2x6x13+x7x1434)h5+(2x1x8+2x3x10+2x4x11+2x5x12+2x6x13+2x7x1442)h4+(2x1x8+x3x10+x4x11+2x5x12+x6x13+x7x1428)h3+(x1x8+x3x10+x4x11+x5x12+x6x13+x7x1421)h2+(x1x8+x3x10+x5x12+x7x1414)h1[e,f] - h = \left(x_{2} x_{9} -10\right)h_{8}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{6} x_{13} +x_{7} x_{14} -18\right)h_{7}+\left(2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -26\right)h_{6}+\left(2x_{1} x_{8} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -34\right)h_{5}+\left(2x_{1} x_{8} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -42\right)h_{4}+\left(2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -28\right)h_{3}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -21\right)h_{2}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -14\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x3x10+x5x12+x7x1414=0x1x8+x3x10+x4x11+x5x12+x6x13+x7x1421=02x1x8+x3x10+x4x11+2x5x12+x6x13+x7x1428=02x1x8+2x3x10+2x4x11+2x5x12+2x6x13+2x7x1442=02x1x8+2x3x10+x4x11+x5x12+2x6x13+x7x1434=02x1x8+x3x10+x4x11+x6x13+x7x1426=0x1x8+x2x9+x6x13+x7x1418=0x2x910=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -14&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -21&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -28&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -42&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -34&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -26&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{6} x_{13} +x_{7} x_{14} -18&=&0\\x_{2} x_{9} -10&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+18h7+26h6+34h5+42h4+28h3+21h2+14h1e=x1g71+x6g54+x7g53+x3g52+x5g44+x4g39+x2g15f=g15+g39+g44+g52+g53+g54+g71\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\\e&=&x_{1} g_{71}+x_{6} g_{54}+x_{7} g_{53}+x_{3} g_{52}+x_{5} g_{44}+x_{4} g_{39}+x_{2} g_{15}\\f&=&g_{-15}+g_{-39}+g_{-44}+g_{-52}+g_{-53}+g_{-54}+g_{-71}\end{array}
Matrix form of the system we are trying to solve: (10101011011111201121120222222021121201101111000110100000)[col. vect.]=(1421284234261810)\begin{pmatrix}1 & 0 & 1 & 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 1 & 1 & 1 & 1\\ 2 & 0 & 1 & 1 & 2 & 1 & 1\\ 2 & 0 & 2 & 2 & 2 & 2 & 2\\ 2 & 0 & 2 & 1 & 1 & 2 & 1\\ 2 & 0 & 1 & 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 21\\ 28\\ 42\\ 34\\ 26\\ 18\\ 10\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+18h7+26h6+34h5+42h4+28h3+21h2+14h1e=x1g71+x6g54+x7g53+x3g52+x5g44+x4g39+x2g15f=x9g15+x11g39+x12g44+x10g52+x14g53+x13g54+x8g71\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\\ e&=&x_{1} g_{71}+x_{6} g_{54}+x_{7} g_{53}+x_{3} g_{52}+x_{5} g_{44}+x_{4} g_{39}+x_{2} g_{15}\\ f&=&x_{9} g_{-15}+x_{11} g_{-39}+x_{12} g_{-44}+x_{10} g_{-52}+x_{14} g_{-53}+x_{13} g_{-54}+x_{8} g_{-71}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x3x10+x5x12+x7x1414=0x1x8+x3x10+x4x11+x5x12+x6x13+x7x1421=02x1x8+x3x10+x4x11+2x5x12+x6x13+x7x1428=02x1x8+2x3x10+2x4x11+2x5x12+2x6x13+2x7x1442=02x1x8+2x3x10+x4x11+x5x12+2x6x13+x7x1434=02x1x8+x3x10+x4x11+x6x13+x7x1426=0x1x8+x2x9+x6x13+x7x1418=0x2x910=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -14&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -21&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -28&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -42&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -34&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -26&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{6} x_{13} +x_{7} x_{14} -18&=&0\\x_{2} x_{9} -10&=&0\\\end{array}

A130A^{30}_1
h-characteristic: (1, 0, 0, 0, 0, 1, 0, 2)
Length of the weight dual to h: 60
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: D4+2A1D^{1}_4+2A^{1}_1 Containing regular semisimple subalgebra number 2: D5D^{1}_5
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V10ψ+V8ψ+8V7ψ+8V6ψ+8V5ψ+V4ψ+8V2ψ+8Vψ+15V0V_{10\psi}+V_{8\psi}+8V_{7\psi}+8V_{6\psi}+8V_{5\psi}+V_{4\psi}+8V_{2\psi}+8V_{\psi}+15V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+18h7+26h6+33h5+40h4+27h3+20h2+14h1e=6g75+g63+6g60+6g40+g38+10g8f=g8+g38+g40+g60+g63+g75\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+33h_{5}+40h_{4}+27h_{3}+20h_{2}+14h_{1}\\ e&=&6g_{75}+g_{63}+6g_{60}+6g_{40}+g_{38}+10g_{8}\\ f&=&g_{-8}+g_{-38}+g_{-40}+g_{-60}+g_{-63}+g_{-75}\end{array}
Lie brackets of the above elements.
h=10h8+18h7+26h6+33h5+40h4+27h3+20h2+14h1e=6g75+g63+6g60+6g40+g38+10g8f=g8+g38+g40+g60+g63+g75\begin{array}{rcl}[e, f]&=&10h_{8}+18h_{7}+26h_{6}+33h_{5}+40h_{4}+27h_{3}+20h_{2}+14h_{1}\\ [h, e]&=&12g_{75}+2g_{63}+12g_{60}+12g_{40}+2g_{38}+20g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-38}-2g_{-40}-2g_{-60}-2g_{-63}-2g_{-75}\end{array}
Centralizer type: A3A_3
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Unknown elements.
h=10h8+18h7+26h6+33h5+40h4+27h3+20h2+14h1e=x1g75+x5g63+x3g60+x4g40+x6g38+x2g8e=x8g8+x12g38+x10g40+x9g60+x11g63+x7g75\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+33h_{5}+40h_{4}+27h_{3}+20h_{2}+14h_{1}\\ e&=&x_{1} g_{75}+x_{5} g_{63}+x_{3} g_{60}+x_{4} g_{40}+x_{6} g_{38}+x_{2} g_{8}\\ f&=&x_{8} g_{-8}+x_{12} g_{-38}+x_{10} g_{-40}+x_{9} g_{-60}+x_{11} g_{-63}+x_{7} g_{-75}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x810)h8+(x1x7+x3x9+x4x1018)h7+(x1x7+2x3x9+x4x10+x5x11+x6x1226)h6+(2x1x7+2x3x9+x4x10+2x5x11+x6x1233)h5+(3x1x7+2x3x9+x4x10+3x5x11+x6x1240)h4+(2x1x7+x3x9+x4x10+2x5x11+x6x1227)h3+(2x1x7+x3x9+x5x11+x6x1220)h2+(x1x7+x4x10+x5x11+x6x1214)h1[e,f] - h = \left(x_{2} x_{8} -10\right)h_{8}+\left(x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} -18\right)h_{7}+\left(x_{1} x_{7} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -26\right)h_{6}+\left(2x_{1} x_{7} +2x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -33\right)h_{5}+\left(3x_{1} x_{7} +2x_{3} x_{9} +x_{4} x_{10} +3x_{5} x_{11} +x_{6} x_{12} -40\right)h_{4}+\left(2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -27\right)h_{3}+\left(2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -20\right)h_{2}+\left(x_{1} x_{7} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -14\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x7+x4x10+x5x11+x6x1214=02x1x7+x3x9+x5x11+x6x1220=02x1x7+x3x9+x4x10+2x5x11+x6x1227=03x1x7+2x3x9+x4x10+3x5x11+x6x1240=02x1x7+2x3x9+x4x10+2x5x11+x6x1233=0x1x7+2x3x9+x4x10+x5x11+x6x1226=0x1x7+x3x9+x4x1018=0x2x810=0\begin{array}{rcl}x_{1} x_{7} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -14&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -27&=&0\\3x_{1} x_{7} +2x_{3} x_{9} +x_{4} x_{10} +3x_{5} x_{11} +x_{6} x_{12} -40&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -33&=&0\\x_{1} x_{7} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -26&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} -18&=&0\\x_{2} x_{8} -10&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+18h7+26h6+33h5+40h4+27h3+20h2+14h1e=x1g75+x5g63+x3g60+x4g40+x6g38+x2g8f=g8+g38+g40+g60+g63+g75\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+33h_{5}+40h_{4}+27h_{3}+20h_{2}+14h_{1}\\e&=&x_{1} g_{75}+x_{5} g_{63}+x_{3} g_{60}+x_{4} g_{40}+x_{6} g_{38}+x_{2} g_{8}\\f&=&g_{-8}+g_{-38}+g_{-40}+g_{-60}+g_{-63}+g_{-75}\end{array}
Matrix form of the system we are trying to solve: (100111201011201121302131202121102111101100010000)[col. vect.]=(1420274033261810)\begin{pmatrix}1 & 0 & 0 & 1 & 1 & 1\\ 2 & 0 & 1 & 0 & 1 & 1\\ 2 & 0 & 1 & 1 & 2 & 1\\ 3 & 0 & 2 & 1 & 3 & 1\\ 2 & 0 & 2 & 1 & 2 & 1\\ 1 & 0 & 2 & 1 & 1 & 1\\ 1 & 0 & 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 20\\ 27\\ 40\\ 33\\ 26\\ 18\\ 10\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+18h7+26h6+33h5+40h4+27h3+20h2+14h1e=x1g75+x5g63+x3g60+x4g40+x6g38+x2g8f=x8g8+x12g38+x10g40+x9g60+x11g63+x7g75\begin{array}{rcl}h&=&10h_{8}+18h_{7}+26h_{6}+33h_{5}+40h_{4}+27h_{3}+20h_{2}+14h_{1}\\ e&=&x_{1} g_{75}+x_{5} g_{63}+x_{3} g_{60}+x_{4} g_{40}+x_{6} g_{38}+x_{2} g_{8}\\ f&=&x_{8} g_{-8}+x_{12} g_{-38}+x_{10} g_{-40}+x_{9} g_{-60}+x_{11} g_{-63}+x_{7} g_{-75}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x7+x4x10+x5x11+x6x1214=02x1x7+x3x9+x5x11+x6x1220=02x1x7+x3x9+x4x10+2x5x11+x6x1227=03x1x7+2x3x9+x4x10+3x5x11+x6x1240=02x1x7+2x3x9+x4x10+2x5x11+x6x1233=0x1x7+2x3x9+x4x10+x5x11+x6x1226=0x1x7+x3x9+x4x1018=0x2x810=0\begin{array}{rcl}x_{1} x_{7} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -14&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -27&=&0\\3x_{1} x_{7} +2x_{3} x_{9} +x_{4} x_{10} +3x_{5} x_{11} +x_{6} x_{12} -40&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -33&=&0\\x_{1} x_{7} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -26&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} -18&=&0\\x_{2} x_{8} -10&=&0\\\end{array}

A130A^{30}_1
h-characteristic: (0, 0, 0, 1, 0, 0, 1, 0)
Length of the weight dual to h: 60
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A4+A3A^{1}_4+A^{1}_3
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 2V9ψ+3V8ψ+4V7ψ+6V6ψ+6V5ψ+6V4ψ+8V3ψ+6V2ψ+4Vψ+3V02V_{9\psi}+3V_{8\psi}+4V_{7\psi}+6V_{6\psi}+6V_{5\psi}+6V_{4\psi}+8V_{3\psi}+6V_{2\psi}+4V_{\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=9h8+18h7+26h6+34h5+42h4+28h3+21h2+14h1e=4g57+3g43+6g42+3g41+6g40+4g39+4g37f=g37+g39+g40+g41+g42+g43+g57\begin{array}{rcl}h&=&9h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\\ e&=&4g_{57}+3g_{43}+6g_{42}+3g_{41}+6g_{40}+4g_{39}+4g_{37}\\ f&=&g_{-37}+g_{-39}+g_{-40}+g_{-41}+g_{-42}+g_{-43}+g_{-57}\end{array}
Lie brackets of the above elements.
h=9h8+18h7+26h6+34h5+42h4+28h3+21h2+14h1e=4g57+3g43+6g42+3g41+6g40+4g39+4g37f=g37+g39+g40+g41+g42+g43+g57\begin{array}{rcl}[e, f]&=&9h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\\ [h, e]&=&8g_{57}+6g_{43}+12g_{42}+6g_{41}+12g_{40}+8g_{39}+8g_{37}\\ [h, f]&=&-2g_{-37}-2g_{-39}-2g_{-40}-2g_{-41}-2g_{-42}-2g_{-43}-2g_{-57}\end{array}
Centralizer type: A110A^{10}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 3): h82h5+2h3+h2+2h1h_{8}-2h_{5}+2h_{3}+h_{2}+2h_{1}, g8+2g6+g3g2+2g1+g13g_{8}+2g_{6}+g_{3}-g_{2}+2g_{1}+g_{-13}, g13+12g112g2+g3+12g6+12g8g_{13}+1/2g_{-1}-1/2g_{-2}+g_{-3}+1/2g_{-6}+1/2g_{-8}
Basis of centralizer intersected with cartan (dimension: 1): 12h8+h5h312h2h1-1/2h_{8}+h_{5}-h_{3}-1/2h_{2}-h_{1}
Cartan of centralizer (dimension: 1): 12h8+h5h312h2h1-1/2h_{8}+h_{5}-h_{3}-1/2h_{2}-h_{1}
Cartan-generating semisimple element: 12h8+h5h312h2h1-1/2h_{8}+h_{5}-h_{3}-1/2h_{2}-h_{1}
adjoint action: (000010001)\begin{pmatrix}0 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}
Characteristic polynomial ad H: x3xx^3-x
Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H: 00, 11, 1-1
3 eigenvectors of ad H: 1, 0, 0(1,0,0), 0, 0, 1(0,0,1), 0, 1, 0(0,1,0)
Centralizer type: A^{10}_1
Reductive components (1 total):
Scalar product computed: (1300)\begin{pmatrix}1/300\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h8+2h52h3h22h1-h_{8}+2h_{5}-2h_{3}-h_{2}-2h_{1}
matching e: g13+12g112g2+g3+12g6+12g8g_{13}+1/2g_{-1}-1/2g_{-2}+g_{-3}+1/2g_{-6}+1/2g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h8+2h52h3h22h1-h_{8}+2h_{5}-2h_{3}-h_{2}-2h_{1}
matching e: g13+12g112g2+g3+12g6+12g8g_{13}+1/2g_{-1}-1/2g_{-2}+g_{-3}+1/2g_{-6}+1/2g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (1200)\begin{pmatrix}1200\\ \end{pmatrix}
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Unknown elements.
h=9h8+18h7+26h6+34h5+42h4+28h3+21h2+14h1e=x1g57+x5g43+x2g42+x7g41+x3g40+x4g39+x6g37e=x13g37+x11g39+x10g40+x14g41+x9g42+x12g43+x8g57\begin{array}{rcl}h&=&9h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\\ e&=&x_{1} g_{57}+x_{5} g_{43}+x_{2} g_{42}+x_{7} g_{41}+x_{3} g_{40}+x_{4} g_{39}+x_{6} g_{37}\\ f&=&x_{13} g_{-37}+x_{11} g_{-39}+x_{10} g_{-40}+x_{14} g_{-41}+x_{9} g_{-42}+x_{12} g_{-43}+x_{8} g_{-57}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x9+x5x129)h8+(x2x9+x3x10+x5x12+x7x1418)h7+(x1x8+x2x9+x3x10+x4x11+x5x12+x7x1426)h6+(2x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+x7x1434)h5+(2x1x8+x2x9+x3x10+2x4x11+x5x12+2x6x13+x7x1442)h4+(2x1x8+x3x10+x4x11+x5x12+x6x13+x7x1428)h3+(x1x8+x2x9+x4x11+x6x13+x7x1421)h2+(x1x8+x3x10+x6x1314)h1[e,f] - h = \left(x_{2} x_{9} +x_{5} x_{12} -9\right)h_{8}+\left(x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -18\right)h_{7}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -26\right)h_{6}+\left(2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -34\right)h_{5}+\left(2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -42\right)h_{4}+\left(2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -28\right)h_{3}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -21\right)h_{2}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} -14\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x3x10+x6x1314=0x1x8+x2x9+x4x11+x6x13+x7x1421=02x1x8+x3x10+x4x11+x5x12+x6x13+x7x1428=02x1x8+x2x9+x3x10+2x4x11+x5x12+2x6x13+x7x1442=02x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+x7x1434=0x1x8+x2x9+x3x10+x4x11+x5x12+x7x1426=0x2x9+x3x10+x5x12+x7x1418=0x2x9+x5x129=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} -14&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -21&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -28&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -42&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -34&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -26&=&0\\x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -18&=&0\\x_{2} x_{9} +x_{5} x_{12} -9&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=9h8+18h7+26h6+34h5+42h4+28h3+21h2+14h1e=x1g57+x5g43+x2g42+x7g41+x3g40+x4g39+x6g37f=g37+g39+g40+g41+g42+g43+g57\begin{array}{rcl}h&=&9h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\\e&=&x_{1} g_{57}+x_{5} g_{43}+x_{2} g_{42}+x_{7} g_{41}+x_{3} g_{40}+x_{4} g_{39}+x_{6} g_{37}\\f&=&g_{-37}+g_{-39}+g_{-40}+g_{-41}+g_{-42}+g_{-43}+g_{-57}\end{array}
Matrix form of the system we are trying to solve: (10100101101011201111121121212111111111110101101010100100)[col. vect.]=(142128423426189)\begin{pmatrix}1 & 0 & 1 & 0 & 0 & 1 & 0\\ 1 & 1 & 0 & 1 & 0 & 1 & 1\\ 2 & 0 & 1 & 1 & 1 & 1 & 1\\ 2 & 1 & 1 & 2 & 1 & 2 & 1\\ 2 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 0 & 1\\ 0 & 1 & 1 & 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 21\\ 28\\ 42\\ 34\\ 26\\ 18\\ 9\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=9h8+18h7+26h6+34h5+42h4+28h3+21h2+14h1e=x1g57+x5g43+x2g42+x7g41+x3g40+x4g39+x6g37f=x13g37+x11g39+x10g40+x14g41+x9g42+x12g43+x8g57\begin{array}{rcl}h&=&9h_{8}+18h_{7}+26h_{6}+34h_{5}+42h_{4}+28h_{3}+21h_{2}+14h_{1}\\ e&=&x_{1} g_{57}+x_{5} g_{43}+x_{2} g_{42}+x_{7} g_{41}+x_{3} g_{40}+x_{4} g_{39}+x_{6} g_{37}\\ f&=&x_{13} g_{-37}+x_{11} g_{-39}+x_{10} g_{-40}+x_{14} g_{-41}+x_{9} g_{-42}+x_{12} g_{-43}+x_{8} g_{-57}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x3x10+x6x1314=0x1x8+x2x9+x4x11+x6x13+x7x1421=02x1x8+x3x10+x4x11+x5x12+x6x13+x7x1428=02x1x8+x2x9+x3x10+2x4x11+x5x12+2x6x13+x7x1442=02x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+x7x1434=0x1x8+x2x9+x3x10+x4x11+x5x12+x7x1426=0x2x9+x3x10+x5x12+x7x1418=0x2x9+x5x129=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{6} x_{13} -14&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -21&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -28&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -42&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -34&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -26&=&0\\x_{2} x_{9} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -18&=&0\\x_{2} x_{9} +x_{5} x_{12} -9&=&0\\\end{array}

A129A^{29}_1
h-characteristic: (0, 1, 0, 0, 0, 0, 1, 2)
Length of the weight dual to h: 58
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: D4+A1D^{1}_4+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V10ψ+6V7ψ+14V6ψ+6V5ψ+2V2ψ+14Vψ+21V0V_{10\psi}+6V_{7\psi}+14V_{6\psi}+6V_{5\psi}+2V_{2\psi}+14V_{\psi}+21V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+18h7+25h6+32h5+39h4+26h3+20h2+13h1e=6g81+g69+6g58+6g34+10g8f=g8+g34+g58+g69+g81\begin{array}{rcl}h&=&10h_{8}+18h_{7}+25h_{6}+32h_{5}+39h_{4}+26h_{3}+20h_{2}+13h_{1}\\ e&=&6g_{81}+g_{69}+6g_{58}+6g_{34}+10g_{8}\\ f&=&g_{-8}+g_{-34}+g_{-58}+g_{-69}+g_{-81}\end{array}
Lie brackets of the above elements.
h=10h8+18h7+25h6+32h5+39h4+26h3+20h2+13h1e=6g81+g69+6g58+6g34+10g8f=g8+g34+g58+g69+g81\begin{array}{rcl}[e, f]&=&10h_{8}+18h_{7}+25h_{6}+32h_{5}+39h_{4}+26h_{3}+20h_{2}+13h_{1}\\ [h, e]&=&12g_{81}+2g_{69}+12g_{58}+12g_{34}+20g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-34}-2g_{-58}-2g_{-69}-2g_{-81}\end{array}
Centralizer type: C3C_3
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 21): g6g_{-6}, g4g_{-4}, g1g_{-1}, h1h_{1}, h4h_{4}, h6h_{6}, g1g_{1}, g3+g16g_{3}+g_{-16}, g4g_{4}, g5+g20g_{5}+g_{-20}, g6g_{6}, g9g11g_{9}-g_{-11}, g11g9g_{11}-g_{-9}, g12g13g_{12}-g_{-13}, g13g12g_{13}-g_{-12}, g16+g3g_{16}+g_{-3}, g19+g32g_{19}+g_{-32}, g20+g5g_{20}+g_{-5}, g24g27g_{24}-g_{-27}, g27g24g_{27}-g_{-24}, g32+g19g_{32}+g_{-19}
Basis of centralizer intersected with cartan (dimension: 3): h6-h_{6}, h4-h_{4}, h1-h_{1}
Cartan of centralizer (dimension: 3): h1-h_{1}, h4-h_{4}, h6-h_{6}
Cartan-generating semisimple element: h69h4+7h1-h_{6}-9h_{4}+7h_{1}
adjoint action: (20000000000000000000001800000000000000000000014000000000000000000000000000000000000000000000000000000000000000000000000000000000000000140000000000000000000002000000000000000000000180000000000000000000001000000000000000000000020000000000000000000001600000000000000000000016000000000000000000000800000000000000000000080000000000000000000002000000000000000000000600000000000000000000010000000000000000000000800000000000000000000080000000000000000000006)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6\\ \end{pmatrix}
Characteristic polynomial ad H: x211048x19+434928x1792093696x15+10748645120x13701372504064x11+24564719030272x9405207860641792x7+2228902957154304x53835513169510400x3x^{21}-1048x^{19}+434928x^{17}-92093696x^{15}+10748645120x^{13}-701372504064x^{11}+24564719030272x^9-405207860641792x^7+2228902957154304x^5-3835513169510400x^3
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -18)(x -16)(x -14)(x -10)(x -8)(x -8)(x -6)(x -2)(x -2)(x +2)(x +2)(x +6)(x +8)(x +8)(x +10)(x +14)(x +16)(x +18)
Eigenvalues of ad H: 00, 1818, 1616, 1414, 1010, 88, 66, 22, 2-2, 6-6, 8-8, 10-10, 14-14, 16-16, 18-18
21 eigenvectors of ad H: 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)
Centralizer type: C^{1}_3
Reductive components (1 total):
Scalar product computed: (1601601120160130011200160)\begin{pmatrix}1/60 & -1/60 & -1/120\\ -1/60 & 1/30 & 0\\ -1/120 & 0 & 1/60\\ \end{pmatrix}
Simple basis of Cartan of centralizer (3 total):
h6+h1h_{6}+h_{1}
matching e: g32+g19g_{32}+g_{-19}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000010000000000000000000000000000000000000000000100000000000000000000020000000000000000000001000000000000000000000100000000000000000000010000000000000000000001000000000000000000000100000000000000000000020000000000000000000001000000000000000000000000000000000000000000000000000000000000000002)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
h6-h_{6}
matching e: g6g_{-6}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000020000000000000000000000000000000000000000000000000000000000000000010000000000000000000001000000000000000000000000000000000000000000010000000000000000000001000000000000000000000100000000000000000000010000000000000000000001)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
h4h1-h_{4}-h_{1}
matching e: g3+g16g_{3}+g_{-16}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000020000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000020000000000000000000002000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000001000000000000000000000200000000000000000000010000000000000000000001000000000000000000000100000000000000000000010000000000000000000001)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h6+h1h_{6}+h_{1}
matching e: g32+g19g_{32}+g_{-19}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000010000000000000000000000000000000000000000000100000000000000000000020000000000000000000001000000000000000000000100000000000000000000010000000000000000000001000000000000000000000100000000000000000000020000000000000000000001000000000000000000000000000000000000000000000000000000000000000002)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
h6-h_{6}
matching e: g6g_{-6}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000020000000000000000000000000000000000000000000000000000000000000000010000000000000000000001000000000000000000000000000000000000000000010000000000000000000001000000000000000000000100000000000000000000010000000000000000000001)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
h4h1-h_{4}-h_{1}
matching e: g3+g16g_{3}+g_{-16}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000020000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000020000000000000000000002000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000001000000000000000000000200000000000000000000010000000000000000000001000000000000000000000100000000000000000000010000000000000000000001)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (2, 2, 1), (-2, -2, -1), (1, 1, 0), (-1, -1, 0), (1, 2, 1), (-1, -2, -1), (1, 2, 0), (-1, -2, 0), (1, 0, 1), (-1, 0, -1), (1, 0, 0), (-1, 0, 0), (0, 0, 1), (0, 0, -1), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (24012012012012001200240)\begin{pmatrix}240 & -120 & -120\\ -120 & 120 & 0\\ -120 & 0 & 240\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=10h8+18h7+25h6+32h5+39h4+26h3+20h2+13h1e=x1g81+x5g69+x3g58+x4g34+x2g8e=x7g8+x9g34+x8g58+x10g69+x6g81\begin{array}{rcl}h&=&10h_{8}+18h_{7}+25h_{6}+32h_{5}+39h_{4}+26h_{3}+20h_{2}+13h_{1}\\ e&=&x_{1} g_{81}+x_{5} g_{69}+x_{3} g_{58}+x_{4} g_{34}+x_{2} g_{8}\\ f&=&x_{7} g_{-8}+x_{9} g_{-34}+x_{8} g_{-58}+x_{10} g_{-69}+x_{6} g_{-81}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x710)h8+(x1x6+x3x8+x4x918)h7+(2x1x6+x3x8+x4x9+x5x1025)h6+(3x1x6+x3x8+x4x9+2x5x1032)h5+(3x1x6+2x3x8+x4x9+3x5x1039)h4+(2x1x6+2x3x8+2x5x1026)h3+(x1x6+x3x8+x4x9+2x5x1020)h2+(x1x6+x3x8+x5x1013)h1[e,f] - h = \left(x_{2} x_{7} -10\right)h_{8}+\left(x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} -18\right)h_{7}+\left(2x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -25\right)h_{6}+\left(3x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -32\right)h_{5}+\left(3x_{1} x_{6} +2x_{3} x_{8} +x_{4} x_{9} +3x_{5} x_{10} -39\right)h_{4}+\left(2x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -26\right)h_{3}+\left(x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -20\right)h_{2}+\left(x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -13\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x6+x3x8+x5x1013=0x1x6+x3x8+x4x9+2x5x1020=02x1x6+2x3x8+2x5x1026=03x1x6+2x3x8+x4x9+3x5x1039=03x1x6+x3x8+x4x9+2x5x1032=02x1x6+x3x8+x4x9+x5x1025=0x1x6+x3x8+x4x918=0x2x710=0\begin{array}{rcl}x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -13&=&0\\x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -20&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -26&=&0\\3x_{1} x_{6} +2x_{3} x_{8} +x_{4} x_{9} +3x_{5} x_{10} -39&=&0\\3x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -32&=&0\\2x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -25&=&0\\x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} -18&=&0\\x_{2} x_{7} -10&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+18h7+25h6+32h5+39h4+26h3+20h2+13h1e=x1g81+x5g69+x3g58+x4g34+x2g8f=g8+g34+g58+g69+g81\begin{array}{rcl}h&=&10h_{8}+18h_{7}+25h_{6}+32h_{5}+39h_{4}+26h_{3}+20h_{2}+13h_{1}\\e&=&x_{1} g_{81}+x_{5} g_{69}+x_{3} g_{58}+x_{4} g_{34}+x_{2} g_{8}\\f&=&g_{-8}+g_{-34}+g_{-58}+g_{-69}+g_{-81}\end{array}
Matrix form of the system we are trying to solve: (1010110112202023021330112201111011001000)[col. vect.]=(1320263932251810)\begin{pmatrix}1 & 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 1 & 2\\ 2 & 0 & 2 & 0 & 2\\ 3 & 0 & 2 & 1 & 3\\ 3 & 0 & 1 & 1 & 2\\ 2 & 0 & 1 & 1 & 1\\ 1 & 0 & 1 & 1 & 0\\ 0 & 1 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}13\\ 20\\ 26\\ 39\\ 32\\ 25\\ 18\\ 10\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+18h7+25h6+32h5+39h4+26h3+20h2+13h1e=x1g81+x5g69+x3g58+x4g34+x2g8f=x7g8+x9g34+x8g58+x10g69+x6g81\begin{array}{rcl}h&=&10h_{8}+18h_{7}+25h_{6}+32h_{5}+39h_{4}+26h_{3}+20h_{2}+13h_{1}\\ e&=&x_{1} g_{81}+x_{5} g_{69}+x_{3} g_{58}+x_{4} g_{34}+x_{2} g_{8}\\ f&=&x_{7} g_{-8}+x_{9} g_{-34}+x_{8} g_{-58}+x_{10} g_{-69}+x_{6} g_{-81}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x6+x3x8+x5x1013=0x1x6+x3x8+x4x9+2x5x1020=02x1x6+2x3x8+2x5x1026=03x1x6+2x3x8+x4x9+3x5x1039=03x1x6+x3x8+x4x9+2x5x1032=02x1x6+x3x8+x4x9+x5x1025=0x1x6+x3x8+x4x918=0x2x710=0\begin{array}{rcl}x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -13&=&0\\x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -20&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -26&=&0\\3x_{1} x_{6} +2x_{3} x_{8} +x_{4} x_{9} +3x_{5} x_{10} -39&=&0\\3x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -32&=&0\\2x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -25&=&0\\x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} -18&=&0\\x_{2} x_{7} -10&=&0\\\end{array}

A128A^{28}_1
h-characteristic: (0, 0, 0, 0, 0, 0, 2, 2)
Length of the weight dual to h: 56
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: D4D^{1}_4
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V10ψ+26V6ψ+V2ψ+52V0V_{10\psi}+26V_{6\psi}+V_{2\psi}+52V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=10h8+18h7+24h6+30h5+36h4+24h3+18h2+12h1e=6g97+6g60+10g8+6g7f=g7+g8+g60+g97\begin{array}{rcl}h&=&10h_{8}+18h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ e&=&6g_{97}+6g_{60}+10g_{8}+6g_{7}\\ f&=&g_{-7}+g_{-8}+g_{-60}+g_{-97}\end{array}
Lie brackets of the above elements.
h=10h8+18h7+24h6+30h5+36h4+24h3+18h2+12h1e=6g97+6g60+10g8+6g7f=g7+g8+g60+g97\begin{array}{rcl}[e, f]&=&10h_{8}+18h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ [h, e]&=&12g_{97}+12g_{60}+20g_{8}+12g_{7}\\ [h, f]&=&-2g_{-7}-2g_{-8}-2g_{-60}-2g_{-97}\end{array}
Centralizer type: F4F_4
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 52): g31g_{-31}, g25g_{-25}, g19g_{-19}, g18g_{-18}, g17g_{-17}, g12g_{-12}, g11g_{-11}, g10g_{-10}, g5g_{-5}, g4g_{-4}, g3g_{-3}, g2g_{-2}, h2h_{2}, h3h_{3}, h4h_{4}, h5h_{5}, g1+g44g_{1}+g_{-44}, g2g_{2}, g3g_{3}, g4g_{4}, g5g_{5}, g6g46g_{6}-g_{-46}, g9+g37g_{9}+g_{-37}, g10g_{10}, g11g_{11}, g12g_{12}, g13+g39g_{13}+g_{-39}, g16+g30g_{16}+g_{-30}, g17g_{17}, g18g_{18}, g19g_{19}, g20g33g_{20}-g_{-33}, g23g24g_{23}-g_{-24}, g24g23g_{24}-g_{-23}, g25g_{25}, g26+g27g_{26}+g_{-27}, g27+g26g_{27}+g_{-26}, g30+g16g_{30}+g_{-16}, g31g_{31}, g32g69g_{32}-g_{-69}, g33g20g_{33}-g_{-20}, g37+g9g_{37}+g_{-9}, g38+g63g_{38}+g_{-63}, g39+g13g_{39}+g_{-13}, g44+g1g_{44}+g_{-1}, g45g57g_{45}-g_{-57}, g46g6g_{46}-g_{-6}, g51+g52g_{51}+g_{-52}, g52+g51g_{52}+g_{-51}, g57g45g_{57}-g_{-45}, g63+g38g_{63}+g_{-38}, g69g32g_{69}-g_{-32}
Basis of centralizer intersected with cartan (dimension: 4): h5-h_{5}, h4-h_{4}, h3-h_{3}, h2-h_{2}
Cartan of centralizer (dimension: 4): h4-h_{4}, h3-h_{3}, h2-h_{2}, h5-h_{5}
Cartan-generating semisimple element: h59h4+7h3+5h2-h_{5}-9h_{4}+7h_{3}+5h_{2}
adjoint action: (9000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000000000000000001300000000000000000000000000000000000000000000000000002200000000000000000000000000000000000000000000000000006000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000070000000000000000000000000000000000000000000000000000290000000000000000000000000000000000000000000000000000230000000000000000000000000000000000000000000000000000190000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000007000000000000000000000000000000000000000000000000000019000000000000000000000000000000000000000000000000000023000000000000000000000000000000000000000000000000000029000000000000000000000000000000000000000000000000000070000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000001600000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000006000000000000000000000000000000000000000000000000000022000000000000000000000000000000000000000000000000000080000000000000000000000000000000000000000000000000000130000000000000000000000000000000000000000000000000000130000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000021000000000000000000000000000000000000000000000000000060000000000000000000000000000000000000000000000000000600000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000000130000000000000000000000000000000000000000000000000000900000000000000000000000000000000000000000000000000005000000000000000000000000000000000000000000000000000021000000000000000000000000000000000000000000000000000016000000000000000000000000000000000000000000000000000014000000000000000000000000000000000000000000000000000080000000000000000000000000000000000000000000000000000700000000000000000000000000000000000000000000000000001500000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000080000000000000000000000000000000000000000000000000000800000000000000000000000000000000000000000000000000001500000000000000000000000000000000000000000000000000001400000000000000000000000000000000000000000000000000005)\begin{pmatrix}9 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& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 22 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 29 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 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& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -15 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 15 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -14 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5\\ \end{pmatrix}
Characteristic polynomial ad H: x524590x50+9558675x4812007059540x46+10198626628905x446222399794670690x42+2827534120766651715x40979269678066415494240x38+262389593866288072045995x3654913966359756497473849490x34+9026566278162761914095306033x321168154220572242496004209406660x30+118972089085900647550270860073275x289506310860437312572090009444511710x26+592343232151644844007667867551249145x2428507279876678932222148448774853487560x22+1044653029031316202280743767327990174960x2028553164594674847754233288604172660149760x18+564934908304353667676645150116063323662080x167742424996856345602108019758019089201653760x14+68794856105341489667996472908958043532169216x12358471783174427956339834806838524559859712000x10+949676746188127733288275004115199831572480000x81063603286066708093377701271034218414080000000x6+410808493472109604708193565360626073600000000x4x^{52}-4590x^{50}+9558675x^{48}-12007059540x^{46}+10198626628905x^{44}-6222399794670690x^{42}+2827534120766651715x^{40}-979269678066415494240x^{38}+262389593866288072045995x^{36}-54913966359756497473849490x^{34}+9026566278162761914095306033x^{32}-1168154220572242496004209406660x^{30}+118972089085900647550270860073275x^{28}-9506310860437312572090009444511710x^{26}+592343232151644844007667867551249145x^{24}-28507279876678932222148448774853487560x^{22}+1044653029031316202280743767327990174960x^{20}-28553164594674847754233288604172660149760x^{18}+564934908304353667676645150116063323662080x^{16}-7742424996856345602108019758019089201653760x^{14}+68794856105341489667996472908958043532169216x^{12}-358471783174427956339834806838524559859712000x^{10}+949676746188127733288275004115199831572480000x^8-1063603286066708093377701271034218414080000000x^6+410808493472109604708193565360626073600000000x^4
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -29)(x -23)(x -22)(x -21)(x -20)(x -19)(x -16)(x -15)(x -14)(x -13)(x -13)(x -10)(x -9)(x -8)(x -8)(x -7)(x -7)(x -6)(x -6)(x -5)(x -3)(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(x +3)(x +5)(x +6)(x +6)(x +7)(x +7)(x +8)(x +8)(x +9)(x +10)(x +13)(x +13)(x +14)(x +15)(x +16)(x +19)(x +20)(x +21)(x +22)(x +23)(x +29)
Eigenvalues of ad H: 00, 2929, 2323, 2222, 2121, 2020, 1919, 1616, 1515, 1414, 1313, 1010, 99, 88, 77, 66, 55, 33, 22, 11, 1-1, 2-2, 3-3, 5-5, 6-6, 7-7, 8-8, 9-9, 10-10, 13-13, 14-14, 15-15, 16-16, 19-19, 20-20, 21-21, 22-22, 23-23, 29-29
52 eigenvectors of ad H: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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Centralizer type: F^{1}_4
Reductive components (1 total):
Scalar product computed: (13000160016011201600112016001601600130)\begin{pmatrix}1/30 & 0 & 0 & -1/60\\ 0 & 1/60 & -1/120 & -1/60\\ 0 & -1/120 & 1/60 & 0\\ -1/60 & -1/60 & 0 & 1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (4 total):
h4h3-h_{4}-h_{3}
matching e: g11g_{-11}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000)\begin{pmatrix}1 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2h52h4h3h2-2h_{5}-2h_{4}-h_{3}-h_{2}
matching e: g6g46g_{6}-g_{-46}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h5+2h4+h3+2h2h_{5}+2h_{4}+h_{3}+2h_{2}
matching e: g69g32g_{69}-g_{-32}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h5+h4+h3h_{5}+h_{4}+h_{3}
matching e: g19g_{19}
verification: [h,e]2e=0[h,e]-2e=0
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Linear space basis of intersection of centralizer and ambient Cartan:
h4h3-h_{4}-h_{3}
matching e: g11g_{-11}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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2h52h4h3h2-2h_{5}-2h_{4}-h_{3}-h_{2}
matching e: g6g46g_{6}-g_{-46}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h5+2h4+h3+2h2h_{5}+2h_{4}+h_{3}+2h_{2}
matching e: g69g32g_{69}-g_{-32}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h5+h4+h3h_{5}+h_{4}+h_{3}
matching e: g19g_{19}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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Elements in Cartan dual to root system: (2, 2, 1, 3), (-2, -2, -1, -3), (1, 2, 1, 3), (-1, -2, -1, -3), (1, 2, 1, 2), (-1, -2, -1, -2), (2, 3, 2, 4), (-2, -3, -2, -4), (1, 1, 1, 2), (-1, -1, -1, -2), (1, 1, 1, 1), (-1, -1, -1, -1), (2, 3, 1, 4), (-2, -3, -1, -4), (2, 2, 1, 4), (-2, -2, -1, -4), (2, 2, 1, 2), (-2, -2, -1, -2), (2, 1, 1, 2), (-2, -1, -1, -2), (0, 1, 1, 1), (0, -1, -1, -1), (1, 1, 0, 2), (-1, -1, 0, -2), (1, 1, 0, 1), (-1, -1, 0, -1), (2, 1, 0, 2), (-2, -1, 0, -2), (0, 2, 1, 2), (0, -2, -1, -2), (1, 0, 0, 1), (-1, 0, 0, -1), (0, 1, 1, 2), (0, -1, -1, -2), (1, 0, 0, 0), (-1, 0, 0, 0), (0, 1, 1, 0), (0, -1, -1, 0), (0, 0, 1, 0), (0, 0, -1, 0), (0, 1, 0, 1), (0, -1, 0, -1), (0, 1, 0, 2), (0, -1, 0, -2), (0, 0, 0, 1), (0, 0, 0, -1), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (1200060024012012001202400601200120)\begin{pmatrix}120 & 0 & 0 & -60\\ 0 & 240 & -120 & -120\\ 0 & -120 & 240 & 0\\ -60 & -120 & 0 & 120\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=10h8+18h7+24h6+30h5+36h4+24h3+18h2+12h1e=x1g97+x3g60+x2g8+x4g7e=x8g7+x6g8+x7g60+x5g97\begin{array}{rcl}h&=&10h_{8}+18h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ e&=&x_{1} g_{97}+x_{3} g_{60}+x_{2} g_{8}+x_{4} g_{7}\\ f&=&x_{8} g_{-7}+x_{6} g_{-8}+x_{7} g_{-60}+x_{5} g_{-97}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x610)h8+(x1x5+x3x7+x4x818)h7+(2x1x5+2x3x724)h6+(3x1x5+2x3x730)h5+(4x1x5+2x3x736)h4+(3x1x5+x3x724)h3+(2x1x5+x3x718)h2+(2x1x512)h1[e,f] - h = \left(x_{2} x_{6} -10\right)h_{8}+\left(x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -18\right)h_{7}+\left(2x_{1} x_{5} +2x_{3} x_{7} -24\right)h_{6}+\left(3x_{1} x_{5} +2x_{3} x_{7} -30\right)h_{5}+\left(4x_{1} x_{5} +2x_{3} x_{7} -36\right)h_{4}+\left(3x_{1} x_{5} +x_{3} x_{7} -24\right)h_{3}+\left(2x_{1} x_{5} +x_{3} x_{7} -18\right)h_{2}+\left(2x_{1} x_{5} -12\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
2x1x512=02x1x5+x3x718=03x1x5+x3x724=04x1x5+2x3x736=03x1x5+2x3x730=02x1x5+2x3x724=0x1x5+x3x7+x4x818=0x2x610=0\begin{array}{rcl}2x_{1} x_{5} -12&=&0\\2x_{1} x_{5} +x_{3} x_{7} -18&=&0\\3x_{1} x_{5} +x_{3} x_{7} -24&=&0\\4x_{1} x_{5} +2x_{3} x_{7} -36&=&0\\3x_{1} x_{5} +2x_{3} x_{7} -30&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -24&=&0\\x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -18&=&0\\x_{2} x_{6} -10&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=10h8+18h7+24h6+30h5+36h4+24h3+18h2+12h1e=x1g97+x3g60+x2g8+x4g7f=g7+g8+g60+g97\begin{array}{rcl}h&=&10h_{8}+18h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\e&=&x_{1} g_{97}+x_{3} g_{60}+x_{2} g_{8}+x_{4} g_{7}\\f&=&g_{-7}+g_{-8}+g_{-60}+g_{-97}\end{array}
Matrix form of the system we are trying to solve: (20002010301040203020202010110100)[col. vect.]=(1218243630241810)\begin{pmatrix}2 & 0 & 0 & 0\\ 2 & 0 & 1 & 0\\ 3 & 0 & 1 & 0\\ 4 & 0 & 2 & 0\\ 3 & 0 & 2 & 0\\ 2 & 0 & 2 & 0\\ 1 & 0 & 1 & 1\\ 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}12\\ 18\\ 24\\ 36\\ 30\\ 24\\ 18\\ 10\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=10h8+18h7+24h6+30h5+36h4+24h3+18h2+12h1e=x1g97+x3g60+x2g8+x4g7f=x8g7+x6g8+x7g60+x5g97\begin{array}{rcl}h&=&10h_{8}+18h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ e&=&x_{1} g_{97}+x_{3} g_{60}+x_{2} g_{8}+x_{4} g_{7}\\ f&=&x_{8} g_{-7}+x_{6} g_{-8}+x_{7} g_{-60}+x_{5} g_{-97}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
2x1x512=02x1x5+x3x718=03x1x5+x3x724=04x1x5+2x3x736=03x1x5+2x3x730=02x1x5+2x3x724=0x1x5+x3x7+x4x818=0x2x610=0\begin{array}{rcl}2x_{1} x_{5} -12&=&0\\2x_{1} x_{5} +x_{3} x_{7} -18&=&0\\3x_{1} x_{5} +x_{3} x_{7} -24&=&0\\4x_{1} x_{5} +2x_{3} x_{7} -36&=&0\\3x_{1} x_{5} +2x_{3} x_{7} -30&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -24&=&0\\x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -18&=&0\\x_{2} x_{6} -10&=&0\\\end{array}

A125A^{25}_1
h-characteristic: (0, 0, 1, 0, 0, 1, 0, 0)
Length of the weight dual to h: 50
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A4+A2+A1A^{1}_4+A^{1}_2+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 3V8ψ+4V7ψ+5V6ψ+6V5ψ+10V4ψ+8V3ψ+7V2ψ+6Vψ+3V03V_{8\psi}+4V_{7\psi}+5V_{6\psi}+6V_{5\psi}+10V_{4\psi}+8V_{3\psi}+7V_{2\psi}+6V_{\psi}+3V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=8h8+16h7+24h6+31h5+38h4+26h3+19h2+13h1e=4g67+6g48+2g46+g44+4g43+2g40+6g38f=g38+g40+g43+g44+g46+g48+g67\begin{array}{rcl}h&=&8h_{8}+16h_{7}+24h_{6}+31h_{5}+38h_{4}+26h_{3}+19h_{2}+13h_{1}\\ e&=&4g_{67}+6g_{48}+2g_{46}+g_{44}+4g_{43}+2g_{40}+6g_{38}\\ f&=&g_{-38}+g_{-40}+g_{-43}+g_{-44}+g_{-46}+g_{-48}+g_{-67}\end{array}
Lie brackets of the above elements.
h=8h8+16h7+24h6+31h5+38h4+26h3+19h2+13h1e=4g67+6g48+2g46+g44+4g43+2g40+6g38f=g38+g40+g43+g44+g46+g48+g67\begin{array}{rcl}[e, f]&=&8h_{8}+16h_{7}+24h_{6}+31h_{5}+38h_{4}+26h_{3}+19h_{2}+13h_{1}\\ [h, e]&=&8g_{67}+12g_{48}+4g_{46}+2g_{44}+8g_{43}+4g_{40}+12g_{38}\\ [h, f]&=&-2g_{-38}-2g_{-40}-2g_{-43}-2g_{-44}-2g_{-46}-2g_{-48}-2g_{-67}\end{array}
Centralizer type: A115A^{15}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 3): h7+12h5h4+12h212h1h_{7}+1/2h_{5}-h_{4}+1/2h_{2}-1/2h_{1}, g12+3g102g8+g1g152g18g_{12}+3g_{10}-2g_{8}+g_{1}-g_{-15}-2g_{-18}, g18+g1512g1+12g812g1032g12g_{18}+g_{15}-1/2g_{-1}+1/2g_{-8}-1/2g_{-10}-3/2g_{-12}
Basis of centralizer intersected with cartan (dimension: 1): 2h7+h52h4+h2h12h_{7}+h_{5}-2h_{4}+h_{2}-h_{1}
Cartan of centralizer (dimension: 1): 2h7+h52h4+h2h12h_{7}+h_{5}-2h_{4}+h_{2}-h_{1}
Cartan-generating semisimple element: 2h7+h52h4+h2h12h_{7}+h_{5}-2h_{4}+h_{2}-h_{1}
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Characteristic polynomial ad H: x34xx^3-4x
Factorization of characteristic polynomial of ad H: (x )(x -2)(x +2)
Eigenvalues of ad H: 00, 22, 2-2
3 eigenvectors of ad H: 1, 0, 0(1,0,0), 0, 0, 1(0,0,1), 0, 1, 0(0,1,0)
Centralizer type: A^{15}_1
Reductive components (1 total):
Scalar product computed: (1450)\begin{pmatrix}1/450\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
2h7+h52h4+h2h12h_{7}+h_{5}-2h_{4}+h_{2}-h_{1}
matching e: g18+g1512g1+12g812g1032g12g_{18}+g_{15}-1/2g_{-1}+1/2g_{-8}-1/2g_{-10}-3/2g_{-12}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
2h7+h52h4+h2h12h_{7}+h_{5}-2h_{4}+h_{2}-h_{1}
matching e: g18+g1512g1+12g812g1032g12g_{18}+g_{15}-1/2g_{-1}+1/2g_{-8}-1/2g_{-10}-3/2g_{-12}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000020002)\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (1800)\begin{pmatrix}1800\\ \end{pmatrix}
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Unknown elements.
h=8h8+16h7+24h6+31h5+38h4+26h3+19h2+13h1e=x1g67+x2g48+x6g46+x7g44+x4g43+x5g40+x3g38e=x10g38+x12g40+x11g43+x14g44+x13g46+x9g48+x8g67\begin{array}{rcl}h&=&8h_{8}+16h_{7}+24h_{6}+31h_{5}+38h_{4}+26h_{3}+19h_{2}+13h_{1}\\ e&=&x_{1} g_{67}+x_{2} g_{48}+x_{6} g_{46}+x_{7} g_{44}+x_{4} g_{43}+x_{5} g_{40}+x_{3} g_{38}\\ f&=&x_{10} g_{-38}+x_{12} g_{-40}+x_{11} g_{-43}+x_{14} g_{-44}+x_{13} g_{-46}+x_{9} g_{-48}+x_{8} g_{-67}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x8+x4x118)h8+(x1x8+x2x9+x4x11+x5x1216)h7+(x1x8+x2x9+x3x10+x4x11+x5x12+x6x1324)h6+(2x1x8+x2x9+x3x10+x4x11+x5x12+2x6x13+x7x1431)h5+(2x1x8+2x2x9+x3x10+x4x11+x5x12+2x6x13+2x7x1438)h4+(x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1426)h3+(x1x8+x2x9+x3x10+x6x13+x7x1419)h2+(x1x8+x3x10+x5x12+x7x1413)h1[e,f] - h = \left(x_{1} x_{8} +x_{4} x_{11} -8\right)h_{8}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} -16\right)h_{7}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} -24\right)h_{6}+\left(2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -31\right)h_{5}+\left(2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -38\right)h_{4}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -26\right)h_{3}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -19\right)h_{2}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -13\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x3x10+x5x12+x7x1413=0x1x8+x2x9+x3x10+x6x13+x7x1419=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1426=02x1x8+2x2x9+x3x10+x4x11+x5x12+2x6x13+2x7x1438=02x1x8+x2x9+x3x10+x4x11+x5x12+2x6x13+x7x1431=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x1324=0x1x8+x2x9+x4x11+x5x1216=0x1x8+x4x118=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -13&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -19&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -26&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -38&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -31&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} -24&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} -16&=&0\\x_{1} x_{8} +x_{4} x_{11} -8&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=8h8+16h7+24h6+31h5+38h4+26h3+19h2+13h1e=x1g67+x2g48+x6g46+x7g44+x4g43+x5g40+x3g38f=g38+g40+g43+g44+g46+g48+g67\begin{array}{rcl}h&=&8h_{8}+16h_{7}+24h_{6}+31h_{5}+38h_{4}+26h_{3}+19h_{2}+13h_{1}\\e&=&x_{1} g_{67}+x_{2} g_{48}+x_{6} g_{46}+x_{7} g_{44}+x_{4} g_{43}+x_{5} g_{40}+x_{3} g_{38}\\f&=&g_{-38}+g_{-40}+g_{-43}+g_{-44}+g_{-46}+g_{-48}+g_{-67}\end{array}
Matrix form of the system we are trying to solve: (10101011110011111111222111222111121111111011011001001000)[col. vect.]=(131926383124168)\begin{pmatrix}1 & 0 & 1 & 0 & 1 & 0 & 1\\ 1 & 1 & 1 & 0 & 0 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 2\\ 2 & 2 & 1 & 1 & 1 & 2 & 2\\ 2 & 1 & 1 & 1 & 1 & 2 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 1 & 1 & 0 & 1 & 1 & 0 & 0\\ 1 & 0 & 0 & 1 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}13\\ 19\\ 26\\ 38\\ 31\\ 24\\ 16\\ 8\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=8h8+16h7+24h6+31h5+38h4+26h3+19h2+13h1e=x1g67+x2g48+x6g46+x7g44+x4g43+x5g40+x3g38f=x10g38+x12g40+x11g43+x14g44+x13g46+x9g48+x8g67\begin{array}{rcl}h&=&8h_{8}+16h_{7}+24h_{6}+31h_{5}+38h_{4}+26h_{3}+19h_{2}+13h_{1}\\ e&=&x_{1} g_{67}+x_{2} g_{48}+x_{6} g_{46}+x_{7} g_{44}+x_{4} g_{43}+x_{5} g_{40}+x_{3} g_{38}\\ f&=&x_{10} g_{-38}+x_{12} g_{-40}+x_{11} g_{-43}+x_{14} g_{-44}+x_{13} g_{-46}+x_{9} g_{-48}+x_{8} g_{-67}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x3x10+x5x12+x7x1413=0x1x8+x2x9+x3x10+x6x13+x7x1419=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1426=02x1x8+2x2x9+x3x10+x4x11+x5x12+2x6x13+2x7x1438=02x1x8+x2x9+x3x10+x4x11+x5x12+2x6x13+x7x1431=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x1324=0x1x8+x2x9+x4x11+x5x1216=0x1x8+x4x118=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -13&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -19&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -26&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -38&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -31&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} -24&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} -16&=&0\\x_{1} x_{8} +x_{4} x_{11} -8&=&0\\\end{array}

A124A^{24}_1
h-characteristic: (0, 0, 0, 0, 0, 2, 0, 0)
Length of the weight dual to h: 48
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A4+A2A^{1}_4+A^{1}_2 Containing regular semisimple subalgebra number 2: 2D42D^{1}_4
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 3V8ψ+13V6ψ+14V4ψ+18V2ψ+6V03V_{8\psi}+13V_{6\psi}+14V_{4\psi}+18V_{2\psi}+6V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=8h8+16h7+24h6+30h5+36h4+24h3+18h2+12h1e=4g82+2g48+6g46+6g40+2g38+4g22f=g22+g38+g40+g46+g48+g82\begin{array}{rcl}h&=&8h_{8}+16h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ e&=&4g_{82}+2g_{48}+6g_{46}+6g_{40}+2g_{38}+4g_{22}\\ f&=&g_{-22}+g_{-38}+g_{-40}+g_{-46}+g_{-48}+g_{-82}\end{array}
Lie brackets of the above elements.
h=8h8+16h7+24h6+30h5+36h4+24h3+18h2+12h1e=4g82+2g48+6g46+6g40+2g38+4g22f=g22+g38+g40+g46+g48+g82\begin{array}{rcl}[e, f]&=&8h_{8}+16h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ [h, e]&=&8g_{82}+4g_{48}+12g_{46}+12g_{40}+4g_{38}+8g_{22}\\ [h, f]&=&-2g_{-22}-2g_{-38}-2g_{-40}-2g_{-46}-2g_{-48}-2g_{-82}\end{array}
Centralizer type: A115+A1A^{15}_1+A_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 6): g3g_{-3}, h3h_{3}, h72h4h1h_{7}-2h_{4}-h_{1}, g3g_{3}, g30+g15+12g8+32g23+12g2412g31g_{30}+g_{15}+1/2g_{-8}+3/2g_{-23}+1/2g_{-24}-1/2g_{-31}, g313g24g232g8g152g30g_{31}-3g_{24}-g_{23}-2g_{8}-g_{-15}-2g_{-30}
Basis of centralizer intersected with cartan (dimension: 2): h72h4h1h_{7}-2h_{4}-h_{1}, h3-h_{3}
Cartan of centralizer (dimension: 2): h72h4h1h_{7}-2h_{4}-h_{1}, h3-h_{3}
Cartan-generating semisimple element: h72h4h3h1h_{7}-2h_{4}-h_{3}-h_{1}
adjoint action: (100000000000000000000100000010000001)\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Characteristic polynomial ad H: x62x4+x2x^6-2x^4+x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -1)(x -1)(x +1)(x +1)
Eigenvalues of ad H: 00, 11, 1-1
6 eigenvectors of ad H: 0, 1, 0, 0, 0, 0(0,1,0,0,0,0), 0, 0, 1, 0, 0, 0(0,0,1,0,0,0), 0, 0, 0, 1, 0, 0(0,0,0,1,0,0), 0, 0, 0, 0, 1, 0(0,0,0,0,1,0), 1, 0, 0, 0, 0, 0(1,0,0,0,0,0), 0, 0, 0, 0, 0, 1(0,0,0,0,0,1)
Centralizer type: A^{15}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: (1450)\begin{pmatrix}1/450\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
2h74h43h32h12h_{7}-4h_{4}-3h_{3}-2h_{1}
matching e: g30+g15+12g8+32g23+12g2412g31g_{30}+g_{15}+1/2g_{-8}+3/2g_{-23}+1/2g_{-24}-1/2g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000000020000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
2h74h43h32h12h_{7}-4h_{4}-3h_{3}-2h_{1}
matching e: g30+g15+12g8+32g23+12g2412g31g_{30}+g_{15}+1/2g_{-8}+3/2g_{-23}+1/2g_{-24}-1/2g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000000020000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (1800)\begin{pmatrix}1800\\ \end{pmatrix}

Scalar product computed: (130)\begin{pmatrix}1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h3h_{3}
matching e: g3g_{3}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000200000000000000)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h3h_{3}
matching e: g3g_{3}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000200000000000000)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120)\begin{pmatrix}120\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=8h8+16h7+24h6+30h5+36h4+24h3+18h2+12h1e=x1g82+x5g48+x3g46+x2g40+x6g38+x4g22e=x10g22+x12g38+x8g40+x9g46+x11g48+x7g82\begin{array}{rcl}h&=&8h_{8}+16h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ e&=&x_{1} g_{82}+x_{5} g_{48}+x_{3} g_{46}+x_{2} g_{40}+x_{6} g_{38}+x_{4} g_{22}\\ f&=&x_{10} g_{-22}+x_{12} g_{-38}+x_{8} g_{-40}+x_{9} g_{-46}+x_{11} g_{-48}+x_{7} g_{-82}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x7+x4x108)h8+(x1x7+x2x8+x4x10+x5x1116)h7+(x1x7+x2x8+x3x9+x4x10+x5x11+x6x1224)h6+(2x1x7+x2x8+2x3x9+x5x11+x6x1230)h5+(3x1x7+x2x8+2x3x9+2x5x11+x6x1236)h4+(2x1x7+x2x8+x3x9+x5x11+x6x1224)h3+(2x1x7+x3x9+x5x11+x6x1218)h2+(x1x7+x2x8+x6x1212)h1[e,f] - h = \left(x_{1} x_{7} +x_{4} x_{10} -8\right)h_{8}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{5} x_{11} -16\right)h_{7}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -24\right)h_{6}+\left(2x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -30\right)h_{5}+\left(3x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} +x_{6} x_{12} -36\right)h_{4}+\left(2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -24\right)h_{3}+\left(2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -18\right)h_{2}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{6} x_{12} -12\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x7+x2x8+x6x1212=02x1x7+x3x9+x5x11+x6x1218=02x1x7+x2x8+x3x9+x5x11+x6x1224=03x1x7+x2x8+2x3x9+2x5x11+x6x1236=02x1x7+x2x8+2x3x9+x5x11+x6x1230=0x1x7+x2x8+x3x9+x4x10+x5x11+x6x1224=0x1x7+x2x8+x4x10+x5x1116=0x1x7+x4x108=0\begin{array}{rcl}x_{1} x_{7} +x_{2} x_{8} +x_{6} x_{12} -12&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -18&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -24&=&0\\3x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} +x_{6} x_{12} -36&=&0\\2x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -30&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -24&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{5} x_{11} -16&=&0\\x_{1} x_{7} +x_{4} x_{10} -8&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=8h8+16h7+24h6+30h5+36h4+24h3+18h2+12h1e=x1g82+x5g48+x3g46+x2g40+x6g38+x4g22f=g22+g38+g40+g46+g48+g82\begin{array}{rcl}h&=&8h_{8}+16h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\e&=&x_{1} g_{82}+x_{5} g_{48}+x_{3} g_{46}+x_{2} g_{40}+x_{6} g_{38}+x_{4} g_{22}\\f&=&g_{-22}+g_{-38}+g_{-40}+g_{-46}+g_{-48}+g_{-82}\end{array}
Matrix form of the system we are trying to solve: (110001201011211011312021212011111111110110100100)[col. vect.]=(121824363024168)\begin{pmatrix}1 & 1 & 0 & 0 & 0 & 1\\ 2 & 0 & 1 & 0 & 1 & 1\\ 2 & 1 & 1 & 0 & 1 & 1\\ 3 & 1 & 2 & 0 & 2 & 1\\ 2 & 1 & 2 & 0 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}12\\ 18\\ 24\\ 36\\ 30\\ 24\\ 16\\ 8\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=8h8+16h7+24h6+30h5+36h4+24h3+18h2+12h1e=x1g82+x5g48+x3g46+x2g40+x6g38+x4g22f=x10g22+x12g38+x8g40+x9g46+x11g48+x7g82\begin{array}{rcl}h&=&8h_{8}+16h_{7}+24h_{6}+30h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ e&=&x_{1} g_{82}+x_{5} g_{48}+x_{3} g_{46}+x_{2} g_{40}+x_{6} g_{38}+x_{4} g_{22}\\ f&=&x_{10} g_{-22}+x_{12} g_{-38}+x_{8} g_{-40}+x_{9} g_{-46}+x_{11} g_{-48}+x_{7} g_{-82}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x7+x2x8+x6x1212=02x1x7+x3x9+x5x11+x6x1218=02x1x7+x2x8+x3x9+x5x11+x6x1224=03x1x7+x2x8+2x3x9+2x5x11+x6x1236=02x1x7+x2x8+2x3x9+x5x11+x6x1230=0x1x7+x2x8+x3x9+x4x10+x5x11+x6x1224=0x1x7+x2x8+x4x10+x5x1116=0x1x7+x4x108=0\begin{array}{rcl}x_{1} x_{7} +x_{2} x_{8} +x_{6} x_{12} -12&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -18&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -24&=&0\\3x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} +x_{6} x_{12} -36&=&0\\2x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -30&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -24&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{5} x_{11} -16&=&0\\x_{1} x_{7} +x_{4} x_{10} -8&=&0\\\end{array}

A122A^{22}_1
h-characteristic: (0, 0, 0, 1, 0, 0, 0, 1)
Length of the weight dual to h: 44
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: 2A3+2A12A^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 2: A4+2A1A^{1}_4+2A^{1}_1 Containing regular semisimple subalgebra number 3: D4+A3D^{1}_4+A^{1}_3
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V8ψ+4V7ψ+5V6ψ+8V5ψ+9V4ψ+8V3ψ+9V2ψ+8Vψ+4V0V_{8\psi}+4V_{7\psi}+5V_{6\psi}+8V_{5\psi}+9V_{4\psi}+8V_{3\psi}+9V_{2\psi}+8V_{\psi}+4V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=8h8+15h7+22h6+29h5+36h4+24h3+18h2+12h1e=3g71+3g59+g48+g46+3g45+3g44+4g43+4g42f=g42+g43+g44+g45+g46+g48+g59+g71\begin{array}{rcl}h&=&8h_{8}+15h_{7}+22h_{6}+29h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ e&=&3g_{71}+3g_{59}+g_{48}+g_{46}+3g_{45}+3g_{44}+4g_{43}+4g_{42}\\ f&=&g_{-42}+g_{-43}+g_{-44}+g_{-45}+g_{-46}+g_{-48}+g_{-59}+g_{-71}\end{array}
Lie brackets of the above elements.
h=8h8+15h7+22h6+29h5+36h4+24h3+18h2+12h1e=3g71+3g59+g48+g46+3g45+3g44+4g43+4g42f=g42+g43+g44+g45+g46+g48+g59+g71\begin{array}{rcl}[e, f]&=&8h_{8}+15h_{7}+22h_{6}+29h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ [h, e]&=&6g_{71}+6g_{59}+2g_{48}+2g_{46}+6g_{45}+6g_{44}+8g_{43}+8g_{42}\\ [h, f]&=&-2g_{-42}-2g_{-43}-2g_{-44}-2g_{-45}-2g_{-46}-2g_{-48}-2g_{-59}-2g_{-71}\end{array}
Centralizer type: A12A^{2}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 4): g6+g3g2+g2g3g6g_{6}+g_{3}-g_{2}+g_{-2}-g_{-3}-g_{-6}, g7+g5+g5+g7g_{7}+g_{5}+g_{-5}+g_{-7}, g14g13g13+g14g_{14}-g_{13}-g_{-13}+g_{-14}, g21+g3g2+g2g3g21g_{21}+g_{3}-g_{2}+g_{-2}-g_{-3}-g_{-21}
Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 2): g21+g6+2g32g2+2g22g3g6g21g_{21}+g_{6}+2g_{3}-2g_{2}+2g_{-2}-2g_{-3}-g_{-6}-g_{-21}, g14+g13g7g5g5g7+g13g14-g_{14}+g_{13}-g_{7}-g_{5}-g_{-5}-g_{-7}+g_{-13}-g_{-14}
Cartan-generating semisimple element: g21+g14g13+g7+g6+g5+2g32g2+2g22g3+g5g6+g7g13+g14g21g_{21}+g_{14}-g_{13}+g_{7}+g_{6}+g_{5}+2g_{3}-2g_{2}+2g_{-2}-2g_{-3}+g_{-5}-g_{-6}+g_{-7}-g_{-13}+g_{-14}-g_{-21}
adjoint action: (0220100110010220)\begin{pmatrix}0 & 2 & -2 & 0\\ 1 & 0 & 0 & -1\\ -1 & 0 & 0 & 1\\ 0 & -2 & 2 & 0\\ \end{pmatrix}
Characteristic polynomial ad H: x48x2x^4-8x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x^2-8)
Eigenvalues of ad H: 00, 222\sqrt{2}, 22-2\sqrt{2}
4 eigenvectors of ad H: 0, 1, 1, 0(0,1,1,0), 1, 0, 0, 1(1,0,0,1), -1, -1/2\sqrt{2}, 1/2\sqrt{2}, 1(1,122,122,1), -1, 1/2\sqrt{2}, -1/2\sqrt{2}, 1(1,122,122,1)
Centralizer type: A^{2}_1
Reductive components (1 total):
Scalar product computed: (160)\begin{pmatrix}1/60\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
122g14122g13+122g7+122g5+122g5+122g7122g13+122g141/2\sqrt{2}g_{14}-1/2\sqrt{2}g_{13}+1/2\sqrt{2}g_{7}+1/2\sqrt{2}g_{5}+1/2\sqrt{2}g_{-5}+1/2\sqrt{2}g_{-7}-1/2\sqrt{2}g_{-13}+1/2\sqrt{2}g_{-14}
matching e: g21+122g14122g13122g7g6122g5122g5+g6122g7122g13+122g14g21g_{21}+1/2\sqrt{2}g_{14}-1/2\sqrt{2}g_{13}-1/2\sqrt{2}g_{7}-g_{6}-1/2\sqrt{2}g_{5}-1/2\sqrt{2}g_{-5}+g_{-6}-1/2\sqrt{2}g_{-7}-1/2\sqrt{2}g_{-13}+1/2\sqrt{2}g_{-14}-g_{-21}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (022012200122122001220220)\begin{pmatrix}0 & \sqrt{2} & -\sqrt{2} & 0\\ 1/2\sqrt{2} & 0 & 0 & -1/2\sqrt{2}\\ -1/2\sqrt{2} & 0 & 0 & 1/2\sqrt{2}\\ 0 & -\sqrt{2} & \sqrt{2} & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
122g14122g13+122g7+122g5+122g5+122g7122g13+122g141/2\sqrt{2}g_{14}-1/2\sqrt{2}g_{13}+1/2\sqrt{2}g_{7}+1/2\sqrt{2}g_{5}+1/2\sqrt{2}g_{-5}+1/2\sqrt{2}g_{-7}-1/2\sqrt{2}g_{-13}+1/2\sqrt{2}g_{-14}
matching e: g21+122g14122g13122g7g6122g5122g5+g6122g7122g13+122g14g21g_{21}+1/2\sqrt{2}g_{14}-1/2\sqrt{2}g_{13}-1/2\sqrt{2}g_{7}-g_{6}-1/2\sqrt{2}g_{5}-1/2\sqrt{2}g_{-5}+g_{-6}-1/2\sqrt{2}g_{-7}-1/2\sqrt{2}g_{-13}+1/2\sqrt{2}g_{-14}-g_{-21}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (022012200122122001220220)\begin{pmatrix}0 & \sqrt{2} & -\sqrt{2} & 0\\ 1/2\sqrt{2} & 0 & 0 & -1/2\sqrt{2}\\ -1/2\sqrt{2} & 0 & 0 & 1/2\sqrt{2}\\ 0 & -\sqrt{2} & \sqrt{2} & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240)\begin{pmatrix}240\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=8h8+15h7+22h6+29h5+36h4+24h3+18h2+12h1e=x1g71+x4g59+x8g48+x7g46+x6g45+x3g44+x5g43+x2g42e=x10g42+x13g43+x11g44+x14g45+x15g46+x16g48+x12g59+x9g71\begin{array}{rcl}h&=&8h_{8}+15h_{7}+22h_{6}+29h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ e&=&x_{1} g_{71}+x_{4} g_{59}+x_{8} g_{48}+x_{7} g_{46}+x_{6} g_{45}+x_{3} g_{44}+x_{5} g_{43}+x_{2} g_{42}\\ f&=&x_{10} g_{-42}+x_{13} g_{-43}+x_{11} g_{-44}+x_{14} g_{-45}+x_{15} g_{-46}+x_{16} g_{-48}+x_{12} g_{-59}+x_{9} g_{-71}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x10+x5x138)h8+(x1x9+x2x10+x4x12+x5x13+x8x1615)h7+(2x1x9+x2x10+x4x12+x5x13+x6x14+x7x15+x8x1622)h6+(2x1x9+x2x10+x3x11+2x4x12+x5x13+x6x14+2x7x15+x8x1629)h5+(2x1x9+x2x10+2x3x11+2x4x12+x5x13+2x6x14+2x7x15+2x8x1636)h4+(2x1x9+2x3x11+x4x12+x5x13+x6x14+x7x15+x8x1624)h3+(x1x9+x2x10+x3x11+x4x12+x6x14+x7x15+x8x1618)h2+(x1x9+x3x11+x4x12+x6x1412)h1[e,f] - h = \left(x_{2} x_{10} +x_{5} x_{13} -8\right)h_{8}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{8} x_{16} -15\right)h_{7}+\left(2x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -22\right)h_{6}+\left(2x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +2x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +2x_{7} x_{15} +x_{8} x_{16} -29\right)h_{5}+\left(2x_{1} x_{9} +x_{2} x_{10} +2x_{3} x_{11} +2x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} +2x_{7} x_{15} +2x_{8} x_{16} -36\right)h_{4}+\left(2x_{1} x_{9} +2x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -24\right)h_{3}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -18\right)h_{2}+\left(x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} -12\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x9+x3x11+x4x12+x6x1412=0x1x9+x2x10+x3x11+x4x12+x6x14+x7x15+x8x1618=02x1x9+2x3x11+x4x12+x5x13+x6x14+x7x15+x8x1624=02x1x9+x2x10+2x3x11+2x4x12+x5x13+2x6x14+2x7x15+2x8x1636=02x1x9+x2x10+x3x11+2x4x12+x5x13+x6x14+2x7x15+x8x1629=02x1x9+x2x10+x4x12+x5x13+x6x14+x7x15+x8x1622=0x1x9+x2x10+x4x12+x5x13+x8x1615=0x2x10+x5x138=0\begin{array}{rcl}x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} -12&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -18&=&0\\2x_{1} x_{9} +2x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -24&=&0\\2x_{1} x_{9} +x_{2} x_{10} +2x_{3} x_{11} +2x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} +2x_{7} x_{15} +2x_{8} x_{16} -36&=&0\\2x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +2x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +2x_{7} x_{15} +x_{8} x_{16} -29&=&0\\2x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -22&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{8} x_{16} -15&=&0\\x_{2} x_{10} +x_{5} x_{13} -8&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=8h8+15h7+22h6+29h5+36h4+24h3+18h2+12h1e=x1g71+x4g59+x8g48+x7g46+x6g45+x3g44+x5g43+x2g42f=g42+g43+g44+g45+g46+g48+g59+g71\begin{array}{rcl}h&=&8h_{8}+15h_{7}+22h_{6}+29h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\e&=&x_{1} g_{71}+x_{4} g_{59}+x_{8} g_{48}+x_{7} g_{46}+x_{6} g_{45}+x_{3} g_{44}+x_{5} g_{43}+x_{2} g_{42}\\f&=&g_{-42}+g_{-43}+g_{-44}+g_{-45}+g_{-46}+g_{-48}+g_{-59}+g_{-71}\end{array}
Matrix form of the system we are trying to solve: (1011010011110111202111112122122221121121210111111101100101001000)[col. vect.]=(121824362922158)\begin{pmatrix}1 & 0 & 1 & 1 & 0 & 1 & 0 & 0\\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1\\ 2 & 0 & 2 & 1 & 1 & 1 & 1 & 1\\ 2 & 1 & 2 & 2 & 1 & 2 & 2 & 2\\ 2 & 1 & 1 & 2 & 1 & 1 & 2 & 1\\ 2 & 1 & 0 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}12\\ 18\\ 24\\ 36\\ 29\\ 22\\ 15\\ 8\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=8h8+15h7+22h6+29h5+36h4+24h3+18h2+12h1e=x1g71+x4g59+x8g48+x7g46+x6g45+x3g44+x5g43+x2g42f=x10g42+x13g43+x11g44+x14g45+x15g46+x16g48+x12g59+x9g71\begin{array}{rcl}h&=&8h_{8}+15h_{7}+22h_{6}+29h_{5}+36h_{4}+24h_{3}+18h_{2}+12h_{1}\\ e&=&x_{1} g_{71}+x_{4} g_{59}+x_{8} g_{48}+x_{7} g_{46}+x_{6} g_{45}+x_{3} g_{44}+x_{5} g_{43}+x_{2} g_{42}\\ f&=&x_{10} g_{-42}+x_{13} g_{-43}+x_{11} g_{-44}+x_{14} g_{-45}+x_{15} g_{-46}+x_{16} g_{-48}+x_{12} g_{-59}+x_{9} g_{-71}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x9+x3x11+x4x12+x6x1412=0x1x9+x2x10+x3x11+x4x12+x6x14+x7x15+x8x1618=02x1x9+2x3x11+x4x12+x5x13+x6x14+x7x15+x8x1624=02x1x9+x2x10+2x3x11+2x4x12+x5x13+2x6x14+2x7x15+2x8x1636=02x1x9+x2x10+x3x11+2x4x12+x5x13+x6x14+2x7x15+x8x1629=02x1x9+x2x10+x4x12+x5x13+x6x14+x7x15+x8x1622=0x1x9+x2x10+x4x12+x5x13+x8x1615=0x2x10+x5x138=0\begin{array}{rcl}x_{1} x_{9} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} -12&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -18&=&0\\2x_{1} x_{9} +2x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -24&=&0\\2x_{1} x_{9} +x_{2} x_{10} +2x_{3} x_{11} +2x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} +2x_{7} x_{15} +2x_{8} x_{16} -36&=&0\\2x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +2x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +2x_{7} x_{15} +x_{8} x_{16} -29&=&0\\2x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -22&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +x_{8} x_{16} -15&=&0\\x_{2} x_{10} +x_{5} x_{13} -8&=&0\\\end{array}

A121A^{21}_1
h-characteristic: (1, 0, 0, 0, 0, 1, 0, 1)
Length of the weight dual to h: 42
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 2A3+A12A^{1}_3+A^{1}_1 Containing regular semisimple subalgebra number 2: A4+A1A^{1}_4+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V8ψ+2V7ψ+7V6ψ+8V5ψ+9V4ψ+8V3ψ+8V2ψ+8Vψ+9V0V_{8\psi}+2V_{7\psi}+7V_{6\psi}+8V_{5\psi}+9V_{4\psi}+8V_{3\psi}+8V_{2\psi}+8V_{\psi}+9V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=8h8+15h7+22h6+28h5+34h4+23h3+17h2+12h1e=3g75+g60+3g52+3g51+4g43+4g42+3g40f=g40+g42+g43+g51+g52+g60+g75\begin{array}{rcl}h&=&8h_{8}+15h_{7}+22h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\\ e&=&3g_{75}+g_{60}+3g_{52}+3g_{51}+4g_{43}+4g_{42}+3g_{40}\\ f&=&g_{-40}+g_{-42}+g_{-43}+g_{-51}+g_{-52}+g_{-60}+g_{-75}\end{array}
Lie brackets of the above elements.
h=8h8+15h7+22h6+28h5+34h4+23h3+17h2+12h1e=3g75+g60+3g52+3g51+4g43+4g42+3g40f=g40+g42+g43+g51+g52+g60+g75\begin{array}{rcl}[e, f]&=&8h_{8}+15h_{7}+22h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\\ [h, e]&=&6g_{75}+2g_{60}+6g_{52}+6g_{51}+8g_{43}+8g_{42}+6g_{40}\\ [h, f]&=&-2g_{-40}-2g_{-42}-2g_{-43}-2g_{-51}-2g_{-52}-2g_{-60}-2g_{-75}\end{array}
Centralizer type: A2A_2
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Unknown elements.
h=8h8+15h7+22h6+28h5+34h4+23h3+17h2+12h1e=x1g75+x7g60+x3g52+x4g51+x2g43+x5g42+x6g40e=x13g40+x12g42+x9g43+x11g51+x10g52+x14g60+x8g75\begin{array}{rcl}h&=&8h_{8}+15h_{7}+22h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\\ e&=&x_{1} g_{75}+x_{7} g_{60}+x_{3} g_{52}+x_{4} g_{51}+x_{2} g_{43}+x_{5} g_{42}+x_{6} g_{40}\\ f&=&x_{13} g_{-40}+x_{12} g_{-42}+x_{9} g_{-43}+x_{11} g_{-51}+x_{10} g_{-52}+x_{14} g_{-60}+x_{8} g_{-75}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x9+x5x128)h8+(x1x8+x2x9+x5x12+x6x13+x7x1415)h7+(x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1422)h6+(2x1x8+x2x9+2x3x10+x4x11+x5x12+x6x13+2x7x1428)h5+(3x1x8+x2x9+2x3x10+2x4x11+x5x12+x6x13+2x7x1434)h4+(2x1x8+x2x9+x3x10+2x4x11+x6x13+x7x1423)h3+(2x1x8+x3x10+x4x11+x5x12+x7x1417)h2+(x1x8+x3x10+x4x11+x6x1312)h1[e,f] - h = \left(x_{2} x_{9} +x_{5} x_{12} -8\right)h_{8}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -15\right)h_{7}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -22\right)h_{6}+\left(2x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -28\right)h_{5}+\left(3x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -34\right)h_{4}+\left(2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -23\right)h_{3}+\left(2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -17\right)h_{2}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -12\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x3x10+x4x11+x6x1312=02x1x8+x3x10+x4x11+x5x12+x7x1417=02x1x8+x2x9+x3x10+2x4x11+x6x13+x7x1423=03x1x8+x2x9+2x3x10+2x4x11+x5x12+x6x13+2x7x1434=02x1x8+x2x9+2x3x10+x4x11+x5x12+x6x13+2x7x1428=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1422=0x1x8+x2x9+x5x12+x6x13+x7x1415=0x2x9+x5x128=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -12&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -17&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -23&=&0\\3x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -34&=&0\\2x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -28&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -22&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -15&=&0\\x_{2} x_{9} +x_{5} x_{12} -8&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=8h8+15h7+22h6+28h5+34h4+23h3+17h2+12h1e=x1g75+x7g60+x3g52+x4g51+x2g43+x5g42+x6g40f=g40+g42+g43+g51+g52+g60+g75\begin{array}{rcl}h&=&8h_{8}+15h_{7}+22h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\\e&=&x_{1} g_{75}+x_{7} g_{60}+x_{3} g_{52}+x_{4} g_{51}+x_{2} g_{43}+x_{5} g_{42}+x_{6} g_{40}\\f&=&g_{-40}+g_{-42}+g_{-43}+g_{-51}+g_{-52}+g_{-60}+g_{-75}\end{array}
Matrix form of the system we are trying to solve: (10110102011101211201131221122121112111111211001110100100)[col. vect.]=(121723342822158)\begin{pmatrix}1 & 0 & 1 & 1 & 0 & 1 & 0\\ 2 & 0 & 1 & 1 & 1 & 0 & 1\\ 2 & 1 & 1 & 2 & 0 & 1 & 1\\ 3 & 1 & 2 & 2 & 1 & 1 & 2\\ 2 & 1 & 2 & 1 & 1 & 1 & 2\\ 1 & 1 & 1 & 1 & 1 & 1 & 2\\ 1 & 1 & 0 & 0 & 1 & 1 & 1\\ 0 & 1 & 0 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}12\\ 17\\ 23\\ 34\\ 28\\ 22\\ 15\\ 8\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=8h8+15h7+22h6+28h5+34h4+23h3+17h2+12h1e=x1g75+x7g60+x3g52+x4g51+x2g43+x5g42+x6g40f=x13g40+x12g42+x9g43+x11g51+x10g52+x14g60+x8g75\begin{array}{rcl}h&=&8h_{8}+15h_{7}+22h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\\ e&=&x_{1} g_{75}+x_{7} g_{60}+x_{3} g_{52}+x_{4} g_{51}+x_{2} g_{43}+x_{5} g_{42}+x_{6} g_{40}\\ f&=&x_{13} g_{-40}+x_{12} g_{-42}+x_{9} g_{-43}+x_{11} g_{-51}+x_{10} g_{-52}+x_{14} g_{-60}+x_{8} g_{-75}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x3x10+x4x11+x6x1312=02x1x8+x3x10+x4x11+x5x12+x7x1417=02x1x8+x2x9+x3x10+2x4x11+x6x13+x7x1423=03x1x8+x2x9+2x3x10+2x4x11+x5x12+x6x13+2x7x1434=02x1x8+x2x9+2x3x10+x4x11+x5x12+x6x13+2x7x1428=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1422=0x1x8+x2x9+x5x12+x6x13+x7x1415=0x2x9+x5x128=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -12&=&0\\2x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -17&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -23&=&0\\3x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -34&=&0\\2x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -28&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -22&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -15&=&0\\x_{2} x_{9} +x_{5} x_{12} -8&=&0\\\end{array}

A120A^{20}_1
h-characteristic: (2, 0, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 2A32A^{1}_3 Containing regular semisimple subalgebra number 2: A4A^{1}_4
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V8ψ+11V6ψ+21V4ψ+11V2ψ+24V0V_{8\psi}+11V_{6\psi}+21V_{4\psi}+11V_{2\psi}+24V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=8h8+14h7+20h6+26h5+32h4+22h3+16h2+12h1e=3g93+3g52+4g43+4g42+3g40+3g23f=g23+g40+g42+g43+g52+g93\begin{array}{rcl}h&=&8h_{8}+14h_{7}+20h_{6}+26h_{5}+32h_{4}+22h_{3}+16h_{2}+12h_{1}\\ e&=&3g_{93}+3g_{52}+4g_{43}+4g_{42}+3g_{40}+3g_{23}\\ f&=&g_{-23}+g_{-40}+g_{-42}+g_{-43}+g_{-52}+g_{-93}\end{array}
Lie brackets of the above elements.
h=8h8+14h7+20h6+26h5+32h4+22h3+16h2+12h1e=3g93+3g52+4g43+4g42+3g40+3g23f=g23+g40+g42+g43+g52+g93\begin{array}{rcl}[e, f]&=&8h_{8}+14h_{7}+20h_{6}+26h_{5}+32h_{4}+22h_{3}+16h_{2}+12h_{1}\\ [h, e]&=&6g_{93}+6g_{52}+8g_{43}+8g_{42}+6g_{40}+6g_{23}\\ [h, f]&=&-2g_{-23}-2g_{-40}-2g_{-42}-2g_{-43}-2g_{-52}-2g_{-93}\end{array}
Centralizer type: A4A_4
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Unknown elements.
h=8h8+14h7+20h6+26h5+32h4+22h3+16h2+12h1e=x1g93+x4g52+x5g43+x2g42+x3g40+x6g23e=x12g23+x9g40+x8g42+x11g43+x10g52+x7g93\begin{array}{rcl}h&=&8h_{8}+14h_{7}+20h_{6}+26h_{5}+32h_{4}+22h_{3}+16h_{2}+12h_{1}\\ e&=&x_{1} g_{93}+x_{4} g_{52}+x_{5} g_{43}+x_{2} g_{42}+x_{3} g_{40}+x_{6} g_{23}\\ f&=&x_{12} g_{-23}+x_{9} g_{-40}+x_{8} g_{-42}+x_{11} g_{-43}+x_{10} g_{-52}+x_{7} g_{-93}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x2x8+x5x118)h8+(x1x7+x2x8+x3x9+x5x1114)h7+(2x1x7+x2x8+x3x9+x4x10+x5x1120)h6+(3x1x7+x2x8+x3x9+2x4x10+x5x1126)h5+(4x1x7+x2x8+x3x9+2x4x10+x5x11+x6x1232)h4+(3x1x7+x3x9+x4x10+x5x11+x6x1222)h3+(2x1x7+x2x8+x4x10+x6x1216)h2+(x1x7+x3x9+x4x10+x6x1212)h1[e,f] - h = \left(x_{2} x_{8} +x_{5} x_{11} -8\right)h_{8}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{5} x_{11} -14\right)h_{7}+\left(2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} -20\right)h_{6}+\left(3x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +2x_{4} x_{10} +x_{5} x_{11} -26\right)h_{5}+\left(4x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +2x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -32\right)h_{4}+\left(3x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -22\right)h_{3}+\left(2x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{6} x_{12} -16\right)h_{2}+\left(x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -12\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x7+x3x9+x4x10+x6x1212=02x1x7+x2x8+x4x10+x6x1216=03x1x7+x3x9+x4x10+x5x11+x6x1222=04x1x7+x2x8+x3x9+2x4x10+x5x11+x6x1232=03x1x7+x2x8+x3x9+2x4x10+x5x1126=02x1x7+x2x8+x3x9+x4x10+x5x1120=0x1x7+x2x8+x3x9+x5x1114=0x2x8+x5x118=0\begin{array}{rcl}x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -12&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{6} x_{12} -16&=&0\\3x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -22&=&0\\4x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +2x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -32&=&0\\3x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +2x_{4} x_{10} +x_{5} x_{11} -26&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} -20&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{5} x_{11} -14&=&0\\x_{2} x_{8} +x_{5} x_{11} -8&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=8h8+14h7+20h6+26h5+32h4+22h3+16h2+12h1e=x1g93+x4g52+x5g43+x2g42+x3g40+x6g23f=g23+g40+g42+g43+g52+g93\begin{array}{rcl}h&=&8h_{8}+14h_{7}+20h_{6}+26h_{5}+32h_{4}+22h_{3}+16h_{2}+12h_{1}\\e&=&x_{1} g_{93}+x_{4} g_{52}+x_{5} g_{43}+x_{2} g_{42}+x_{3} g_{40}+x_{6} g_{23}\\f&=&g_{-23}+g_{-40}+g_{-42}+g_{-43}+g_{-52}+g_{-93}\end{array}
Matrix form of the system we are trying to solve: (101101210101301111411211311210211110111010010010)[col. vect.]=(121622322620148)\begin{pmatrix}1 & 0 & 1 & 1 & 0 & 1\\ 2 & 1 & 0 & 1 & 0 & 1\\ 3 & 0 & 1 & 1 & 1 & 1\\ 4 & 1 & 1 & 2 & 1 & 1\\ 3 & 1 & 1 & 2 & 1 & 0\\ 2 & 1 & 1 & 1 & 1 & 0\\ 1 & 1 & 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}12\\ 16\\ 22\\ 32\\ 26\\ 20\\ 14\\ 8\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=8h8+14h7+20h6+26h5+32h4+22h3+16h2+12h1e=x1g93+x4g52+x5g43+x2g42+x3g40+x6g23f=x12g23+x9g40+x8g42+x11g43+x10g52+x7g93\begin{array}{rcl}h&=&8h_{8}+14h_{7}+20h_{6}+26h_{5}+32h_{4}+22h_{3}+16h_{2}+12h_{1}\\ e&=&x_{1} g_{93}+x_{4} g_{52}+x_{5} g_{43}+x_{2} g_{42}+x_{3} g_{40}+x_{6} g_{23}\\ f&=&x_{12} g_{-23}+x_{9} g_{-40}+x_{8} g_{-42}+x_{11} g_{-43}+x_{10} g_{-52}+x_{7} g_{-93}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x7+x3x9+x4x10+x6x1212=02x1x7+x2x8+x4x10+x6x1216=03x1x7+x3x9+x4x10+x5x11+x6x1222=04x1x7+x2x8+x3x9+2x4x10+x5x11+x6x1232=03x1x7+x2x8+x3x9+2x4x10+x5x1126=02x1x7+x2x8+x3x9+x4x10+x5x1120=0x1x7+x2x8+x3x9+x5x1114=0x2x8+x5x118=0\begin{array}{rcl}x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -12&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{6} x_{12} -16&=&0\\3x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -22&=&0\\4x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +2x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -32&=&0\\3x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +2x_{4} x_{10} +x_{5} x_{11} -26&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} -20&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{5} x_{11} -14&=&0\\x_{2} x_{8} +x_{5} x_{11} -8&=&0\\\end{array}

A120A^{20}_1
h-characteristic: (1, 0, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: 2A32A^{1}_3
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 4V7ψ+6V6ψ+4V5ψ+10V4ψ+16V3ψ+6V2ψ+4Vψ+10V04V_{7\psi}+6V_{6\psi}+4V_{5\psi}+10V_{4\psi}+16V_{3\psi}+6V_{2\psi}+4V_{\psi}+10V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=7h8+14h7+21h6+28h5+34h4+23h3+17h2+12h1e=3g66+4g62+4g60+3g47+3g37+3g32f=g32+g37+g47+g60+g62+g66\begin{array}{rcl}h&=&7h_{8}+14h_{7}+21h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\\ e&=&3g_{66}+4g_{62}+4g_{60}+3g_{47}+3g_{37}+3g_{32}\\ f&=&g_{-32}+g_{-37}+g_{-47}+g_{-60}+g_{-62}+g_{-66}\end{array}
Lie brackets of the above elements.
h=7h8+14h7+21h6+28h5+34h4+23h3+17h2+12h1e=3g66+4g62+4g60+3g47+3g37+3g32f=g32+g37+g47+g60+g62+g66\begin{array}{rcl}[e, f]&=&7h_{8}+14h_{7}+21h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\\ [h, e]&=&6g_{66}+8g_{62}+8g_{60}+6g_{47}+6g_{37}+6g_{32}\\ [h, f]&=&-2g_{-32}-2g_{-37}-2g_{-47}-2g_{-60}-2g_{-62}-2g_{-66}\end{array}
Centralizer type: B22B^{2}_2
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 10): h7h2h_{7}-h_{2}, h8+h6h3+h2h_{8}+h_{6}-h_{3}+h_{2}, g2+g7g_{2}+g_{-7}, g3+g22g_{3}+g_{-22}, g7+g2g_{7}+g_{-2}, g10g8g6g11g_{10}-g_{8}-g_{6}-g_{-11}, g11+g6+g8g10g_{11}+g_{-6}+g_{-8}-g_{-10}, g15g14g4g17g_{15}-g_{14}-g_{4}-g_{-17}, g17+g4+g14g15g_{17}+g_{-4}+g_{-14}-g_{-15}, g22+g3g_{22}+g_{-3}
Basis of centralizer intersected with cartan (dimension: 2): h8+h7+h6h3h_{8}+h_{7}+h_{6}-h_{3}, h7h2h_{7}-h_{2}
Cartan of centralizer (dimension: 2): h8+h7+h6h3h_{8}+h_{7}+h_{6}-h_{3}, h7h2h_{7}-h_{2}
Cartan-generating semisimple element: h8+12h7+h6h311h2h_{8}+12h_{7}+h_{6}-h_{3}-11h_{2}
adjoint action: (0000000000000000000000220000000000200000000002200000000001000000000001000000000001200000000001200000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -22 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 22 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 12 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -12 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
Characteristic polynomial ad H: x10732x8+135408x67499584x4+27878400x2x^{10}-732x^8+135408x^6-7499584x^4+27878400x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -22)(x -12)(x -10)(x -2)(x +2)(x +10)(x +12)(x +22)
Eigenvalues of ad H: 00, 2222, 1212, 1010, 22, 2-2, 10-10, 12-12, 22-22
10 eigenvectors of ad H: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,1,0,0), 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,1), 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,1,0), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0)
Centralizer type: B^{2}_2
Reductive components (1 total):
Scalar product computed: (112011201120160)\begin{pmatrix}1/120 & -1/120\\ -1/120 & 1/60\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
h8h6+h3h2-h_{8}-h_{6}+h_{3}-h_{2}
matching e: g11+g6+g8g10g_{11}+g_{-6}+g_{-8}-g_{-10}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000200000000002000000000020000000000200000000002000000000000000000000000000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
h8+h7+h6h3h_{8}+h_{7}+h_{6}-h_{3}
matching e: g22+g3g_{22}+g_{-3}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000000002000000000000000000000100000000001000000000010000000000100000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h8h6+h3h2-h_{8}-h_{6}+h_{3}-h_{2}
matching e: g11+g6+g8g10g_{11}+g_{-6}+g_{-8}-g_{-10}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000200000000002000000000020000000000200000000002000000000000000000000000000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
h8+h7+h6h3h_{8}+h_{7}+h_{6}-h_{3}
matching e: g22+g3g_{22}+g_{-3}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000000002000000000000000000000100000000001000000000010000000000100000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (480240240240)\begin{pmatrix}480 & -240\\ -240 & 240\\ \end{pmatrix}
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Unknown elements.
h=7h8+14h7+21h6+28h5+34h4+23h3+17h2+12h1e=x1g66+x5g62+x2g60+x4g47+x3g37+x6g32e=x12g32+x9g37+x10g47+x8g60+x11g62+x7g66\begin{array}{rcl}h&=&7h_{8}+14h_{7}+21h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\\ e&=&x_{1} g_{66}+x_{5} g_{62}+x_{2} g_{60}+x_{4} g_{47}+x_{3} g_{37}+x_{6} g_{32}\\ f&=&x_{12} g_{-32}+x_{9} g_{-37}+x_{10} g_{-47}+x_{8} g_{-60}+x_{11} g_{-62}+x_{7} g_{-66}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x7+x5x117)h8+(x1x7+x2x8+x4x10+x5x1114)h7+(x1x7+2x2x8+x4x10+x5x11+x6x1221)h6+(x1x7+2x2x8+x3x9+x4x10+2x5x11+x6x1228)h5+(2x1x7+2x2x8+2x3x9+x4x10+2x5x11+x6x1234)h4+(2x1x7+x2x8+x3x9+x4x10+x5x11+x6x1223)h3+(x1x7+x2x8+x3x9+x4x10+x5x1117)h2+(x1x7+x3x9+x4x10+x6x1212)h1[e,f] - h = \left(x_{1} x_{7} +x_{5} x_{11} -7\right)h_{8}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{5} x_{11} -14\right)h_{7}+\left(x_{1} x_{7} +2x_{2} x_{8} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -21\right)h_{6}+\left(x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -28\right)h_{5}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -34\right)h_{4}+\left(2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -23\right)h_{3}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} -17\right)h_{2}+\left(x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -12\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x7+x3x9+x4x10+x6x1212=0x1x7+x2x8+x3x9+x4x10+x5x1117=02x1x7+x2x8+x3x9+x4x10+x5x11+x6x1223=02x1x7+2x2x8+2x3x9+x4x10+2x5x11+x6x1234=0x1x7+2x2x8+x3x9+x4x10+2x5x11+x6x1228=0x1x7+2x2x8+x4x10+x5x11+x6x1221=0x1x7+x2x8+x4x10+x5x1114=0x1x7+x5x117=0\begin{array}{rcl}x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -12&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} -17&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -23&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -34&=&0\\x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -28&=&0\\x_{1} x_{7} +2x_{2} x_{8} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -21&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{5} x_{11} -14&=&0\\x_{1} x_{7} +x_{5} x_{11} -7&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=7h8+14h7+21h6+28h5+34h4+23h3+17h2+12h1e=x1g66+x5g62+x2g60+x4g47+x3g37+x6g32f=g32+g37+g47+g60+g62+g66\begin{array}{rcl}h&=&7h_{8}+14h_{7}+21h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\\e&=&x_{1} g_{66}+x_{5} g_{62}+x_{2} g_{60}+x_{4} g_{47}+x_{3} g_{37}+x_{6} g_{32}\\f&=&g_{-32}+g_{-37}+g_{-47}+g_{-60}+g_{-62}+g_{-66}\end{array}
Matrix form of the system we are trying to solve: (101101111110211111222121121121120111110110100010)[col. vect.]=(121723342821147)\begin{pmatrix}1 & 0 & 1 & 1 & 0 & 1\\ 1 & 1 & 1 & 1 & 1 & 0\\ 2 & 1 & 1 & 1 & 1 & 1\\ 2 & 2 & 2 & 1 & 2 & 1\\ 1 & 2 & 1 & 1 & 2 & 1\\ 1 & 2 & 0 & 1 & 1 & 1\\ 1 & 1 & 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 0 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}12\\ 17\\ 23\\ 34\\ 28\\ 21\\ 14\\ 7\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=7h8+14h7+21h6+28h5+34h4+23h3+17h2+12h1e=x1g66+x5g62+x2g60+x4g47+x3g37+x6g32f=x12g32+x9g37+x10g47+x8g60+x11g62+x7g66\begin{array}{rcl}h&=&7h_{8}+14h_{7}+21h_{6}+28h_{5}+34h_{4}+23h_{3}+17h_{2}+12h_{1}\\ e&=&x_{1} g_{66}+x_{5} g_{62}+x_{2} g_{60}+x_{4} g_{47}+x_{3} g_{37}+x_{6} g_{32}\\ f&=&x_{12} g_{-32}+x_{9} g_{-37}+x_{10} g_{-47}+x_{8} g_{-60}+x_{11} g_{-62}+x_{7} g_{-66}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x7+x3x9+x4x10+x6x1212=0x1x7+x2x8+x3x9+x4x10+x5x1117=02x1x7+x2x8+x3x9+x4x10+x5x11+x6x1223=02x1x7+2x2x8+2x3x9+x4x10+2x5x11+x6x1234=0x1x7+2x2x8+x3x9+x4x10+2x5x11+x6x1228=0x1x7+2x2x8+x4x10+x5x11+x6x1221=0x1x7+x2x8+x4x10+x5x1114=0x1x7+x5x117=0\begin{array}{rcl}x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -12&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} -17&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -23&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -34&=&0\\x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -28&=&0\\x_{1} x_{7} +2x_{2} x_{8} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -21&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{5} x_{11} -14&=&0\\x_{1} x_{7} +x_{5} x_{11} -7&=&0\\\end{array}

A116A^{16}_1
h-characteristic: (0, 2, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 32
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1: D4+A2D^{1}_4+A^{1}_2 Containing regular semisimple subalgebra number 2: D4+4A1D^{1}_4+4A^{1}_1 Containing regular semisimple subalgebra number 3: 4A24A^{1}_2 Containing regular semisimple subalgebra number 4: A3+A2+2A1A^{1}_3+A^{1}_2+2A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 8V6ψ+20V4ψ+28V2ψ+8V08V_{6\psi}+20V_{4\psi}+28V_{2\psi}+8V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=6h8+12h7+18h6+24h5+30h4+20h3+16h2+10h1e=85g9645g92g617g51+4g44+23g41+115g2625g18f=g18+2g26+3g41+g442g613g92+g96\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&8/5g_{96}-4/5g_{92}-g_{61}-7g_{51}+4g_{44}+2/3g_{41}+11/5g_{26}-2/5g_{18}\\ f&=&g_{-18}+2g_{-26}+3g_{-41}+g_{-44}-2g_{-61}-3g_{-92}+g_{-96}\end{array}
Lie brackets of the above elements.
h=6h8+12h7+18h6+24h5+30h4+20h3+16h2+10h1e=85g9645g92g617g51+4g44+23g41+115g2625g18f=g18+2g26+3g41+g442g613g92+g96\begin{array}{rcl}[e, f]&=&6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&16/5g_{96}-8/5g_{92}-2g_{61}-14g_{51}+8g_{44}+4/3g_{41}+22/5g_{26}-4/5g_{18}\\ [h, f]&=&-2g_{-18}-4g_{-26}-6g_{-41}-2g_{-44}+4g_{-61}+6g_{-92}-2g_{-96}\end{array}
Centralizer type: A26A^{6}_2
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 8): h4h1h_{4}-h_{1}, h8h1h_{8}-h_{1}, g1625g14215g715g3+45g15+25g22g_{16}-2/5g_{14}-2/15g_{7}-1/5g_{-3}+4/5g_{-15}+2/5g_{-22}, g22+13g1553g32g7g14+13g16g_{22}+1/3g_{15}-5/3g_{3}-2g_{-7}-g_{-14}+1/3g_{-16}, g32211g24233g21+166g19+133g27533g36g_{32}-2/11g_{24}-2/33g_{21}+1/66g_{-19}+1/33g_{-27}-5/33g_{-36}, g352110g20+65g12+54g5+12g49g_{35}-21/10g_{20}+6/5g_{12}+5/4g_{-5}+1/2g_{-49}, g3611g27+2g19+52g2116g2413g32g_{36}-11g_{27}+2g_{19}+5/2g_{-21}-1/6g_{-24}-1/3g_{-32}, g4975g13+45g5+56g12+12g35g_{49}-7/5g_{13}+4/5g_{5}+5/6g_{-12}+1/2g_{-35}
Basis of centralizer intersected with cartan (dimension: 2): h8h4h_{8}-h_{4}, h8h1h_{8}-h_{1}
Cartan of centralizer (dimension: 2): h8h4h_{8}-h_{4}, h8h1h_{8}-h_{1}
Cartan-generating semisimple element: 2h8h4h12h_{8}-h_{4}-h_{1}
adjoint action: (0000000000000000002000000002000000001000000001000000001000000001)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
Characteristic polynomial ad H: x86x6+9x44x2x^8-6x^6+9x^4-4x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)
Eigenvalues of ad H: 00, 22, 11, 1-1, 2-2
8 eigenvectors of ad H: 1, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0), 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,1), 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,1,0,0), 0, 0, 1, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0)
Centralizer type: A^{6}_2
Reductive components (1 total):
Scalar product computed: (1180136013601180)\begin{pmatrix}1/180 & -1/360\\ -1/360 & 1/180\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
h82h4+h1h_{8}-2h_{4}+h_{1}
matching e: g4975g13+45g5+56g12+12g35g_{49}-7/5g_{13}+4/5g_{5}+5/6g_{-12}+1/2g_{-35}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000001000000001000000001000000002000000001000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
h8+h42h1h_{8}+h_{4}-2h_{1}
matching e: g3611g27+2g19+52g2116g2413g32g_{36}-11g_{27}+2g_{19}+5/2g_{-21}-1/6g_{-24}-1/3g_{-32}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000001000000001000000002000000001000000002000000001)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h82h4+h1h_{8}-2h_{4}+h_{1}
matching e: g4975g13+45g5+56g12+12g35g_{49}-7/5g_{13}+4/5g_{5}+5/6g_{-12}+1/2g_{-35}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000001000000001000000001000000002000000001000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
h8+h42h1h_{8}+h_{4}-2h_{1}
matching e: g3611g27+2g19+52g2116g2413g32g_{36}-11g_{27}+2g_{19}+5/2g_{-21}-1/6g_{-24}-1/3g_{-32}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000001000000001000000002000000001000000002000000001)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (720360360720)\begin{pmatrix}720 & -360\\ -360 & 720\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=6h8+12h7+18h6+24h5+30h4+20h3+16h2+10h1e=x1g96+x6g92+x4g61+x2g51+x7g44+x5g41+x3g26+x8g18e=x16g18+x11g26+x13g41+x15g44+x10g51+x12g61+x14g92+x9g96\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&x_{1} g_{96}+x_{6} g_{92}+x_{4} g_{61}+x_{2} g_{51}+x_{7} g_{44}+x_{5} g_{41}+x_{3} g_{26}+x_{8} g_{18}\\ f&=&x_{16} g_{-18}+x_{11} g_{-26}+x_{13} g_{-41}+x_{15} g_{-44}+x_{10} g_{-51}+x_{12} g_{-61}+x_{14} g_{-92}+x_{9} g_{-96}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x14x2x15x3x16)g6+(x1x9+x4x12+x6x146)h8+(2x1x9+x4x12+x5x13+2x6x1412)h7+(3x1x9+x2x10+x3x11+x4x12+x5x13+2x6x1418)h6+(3x1x9+x2x10+x3x11+x4x12+x5x13+3x6x14+x7x15+x8x1624)h5+(3x1x9+2x2x10+x3x11+2x4x12+x5x13+3x6x14+2x7x15+x8x1630)h4+(2x1x9+2x2x10+x4x12+x5x13+2x6x14+2x7x1520)h3+(x1x9+x2x10+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1616)h2+(x1x9+x2x10+x4x12+x6x14+x7x1510)h1+(x6x9x7x10x8x11)g6[e,f] - h = \left(x_{1} x_{14} -x_{2} x_{15} -x_{3} x_{16} \right)g_{6}+\left(x_{1} x_{9} +x_{4} x_{12} +x_{6} x_{14} -6\right)h_{8}+\left(2x_{1} x_{9} +x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} -12\right)h_{7}+\left(3x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} -18\right)h_{6}+\left(3x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +3x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -24\right)h_{5}+\left(3x_{1} x_{9} +2x_{2} x_{10} +x_{3} x_{11} +2x_{4} x_{12} +x_{5} x_{13} +3x_{6} x_{14} +2x_{7} x_{15} +x_{8} x_{16} -30\right)h_{4}+\left(2x_{1} x_{9} +2x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} +2x_{7} x_{15} -20\right)h_{3}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -16\right)h_{2}+\left(x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} -10\right)h_{1}+\left(x_{6} x_{9} -x_{7} x_{10} -x_{8} x_{11} \right)g_{-6}
The polynomial system that corresponds to finding the h, e, f triple:
x1x9+x2x10+x4x12+x6x14+x7x1510=0x1x9+x2x10+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1616=02x1x9+2x2x10+x4x12+x5x13+2x6x14+2x7x1520=03x1x9+2x2x10+x3x11+2x4x12+x5x13+3x6x14+2x7x15+x8x1630=03x1x9+x2x10+x3x11+x4x12+x5x13+3x6x14+x7x15+x8x1624=03x1x9+x2x10+x3x11+x4x12+x5x13+2x6x1418=02x1x9+x4x12+x5x13+2x6x1412=0x1x9+x4x12+x6x146=0x1x14x2x15x3x16=0x6x9x7x10x8x11=0\begin{array}{rcl}x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} -10&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -16&=&0\\2x_{1} x_{9} +2x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} +2x_{7} x_{15} -20&=&0\\3x_{1} x_{9} +2x_{2} x_{10} +x_{3} x_{11} +2x_{4} x_{12} +x_{5} x_{13} +3x_{6} x_{14} +2x_{7} x_{15} +x_{8} x_{16} -30&=&0\\3x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +3x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -24&=&0\\3x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} -18&=&0\\2x_{1} x_{9} +x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} -12&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{6} x_{14} -6&=&0\\x_{1} x_{14} -x_{2} x_{15} -x_{3} x_{16} &=&0\\x_{6} x_{9} -x_{7} x_{10} -x_{8} x_{11} &=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=6h8+12h7+18h6+24h5+30h4+20h3+16h2+10h1e=x1g96+x6g92+x4g61+x2g51+x7g44+x5g41+x3g26+x8g18f=g18+2g26+3g41+g442g613g92+g96\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+16h_{2}+10h_{1}\\e&=&x_{1} g_{96}+x_{6} g_{92}+x_{4} g_{61}+x_{2} g_{51}+x_{7} g_{44}+x_{5} g_{41}+x_{3} g_{26}+x_{8} g_{18}\\f&=&g_{-18}+2g_{-26}+3g_{-41}+g_{-44}-2g_{-61}-3g_{-92}+g_{-96}\end{array}
Matrix form of the system we are trying to solve: (10020310102233112002362030243921302239113022360020023600100203003110000000000102)[col. vect.]=(10162030241812600)\begin{pmatrix}1 & 0 & 0 & -2 & 0 & -3 & 1 & 0\\ 1 & 0 & 2 & -2 & 3 & -3 & 1 & 1\\ 2 & 0 & 0 & -2 & 3 & -6 & 2 & 0\\ 3 & 0 & 2 & -4 & 3 & -9 & 2 & 1\\ 3 & 0 & 2 & -2 & 3 & -9 & 1 & 1\\ 3 & 0 & 2 & -2 & 3 & -6 & 0 & 0\\ 2 & 0 & 0 & -2 & 3 & -6 & 0 & 0\\ 1 & 0 & 0 & -2 & 0 & -3 & 0 & 0\\ -3 & -1 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & -2\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 20\\ 30\\ 24\\ 18\\ 12\\ 6\\ 0\\ 0\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=6h8+12h7+18h6+24h5+30h4+20h3+16h2+10h1e=x1g96+x6g92+x4g61+x2g51+x7g44+x5g41+x3g26+x8g18f=x16g18+x11g26+x13g41+x15g44+x10g51+x12g61+x14g92+x9g96\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&x_{1} g_{96}+x_{6} g_{92}+x_{4} g_{61}+x_{2} g_{51}+x_{7} g_{44}+x_{5} g_{41}+x_{3} g_{26}+x_{8} g_{18}\\ f&=&x_{16} g_{-18}+x_{11} g_{-26}+x_{13} g_{-41}+x_{15} g_{-44}+x_{10} g_{-51}+x_{12} g_{-61}+x_{14} g_{-92}+x_{9} g_{-96}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x9+x2x10+x4x12+x6x14+x7x1510=0x1x9+x2x10+x3x11+x4x12+x5x13+x6x14+x7x15+x8x1616=02x1x9+2x2x10+x4x12+x5x13+2x6x14+2x7x1520=03x1x9+2x2x10+x3x11+2x4x12+x5x13+3x6x14+2x7x15+x8x1630=03x1x9+x2x10+x3x11+x4x12+x5x13+3x6x14+x7x15+x8x1624=03x1x9+x2x10+x3x11+x4x12+x5x13+2x6x1418=02x1x9+x4x12+x5x13+2x6x1412=0x1x9+x4x12+x6x146=0x1x14x2x15x3x16=0x6x9x7x10x8x11=0\begin{array}{rcl}x_{1} x_{9} +x_{2} x_{10} +x_{4} x_{12} +x_{6} x_{14} +x_{7} x_{15} -10&=&0\\x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -16&=&0\\2x_{1} x_{9} +2x_{2} x_{10} +x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} +2x_{7} x_{15} -20&=&0\\3x_{1} x_{9} +2x_{2} x_{10} +x_{3} x_{11} +2x_{4} x_{12} +x_{5} x_{13} +3x_{6} x_{14} +2x_{7} x_{15} +x_{8} x_{16} -30&=&0\\3x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +3x_{6} x_{14} +x_{7} x_{15} +x_{8} x_{16} -24&=&0\\3x_{1} x_{9} +x_{2} x_{10} +x_{3} x_{11} +x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} -18&=&0\\2x_{1} x_{9} +x_{4} x_{12} +x_{5} x_{13} +2x_{6} x_{14} -12&=&0\\x_{1} x_{9} +x_{4} x_{12} +x_{6} x_{14} -6&=&0\\x_{1} x_{14} -x_{2} x_{15} -x_{3} x_{16} &=&0\\x_{6} x_{9} -x_{7} x_{10} -x_{8} x_{11} &=&0\\\end{array}

A115A^{15}_1
h-characteristic: (0, 0, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 30
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A3+A2+A1A^{1}_3+A^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 2: D4+3A1D^{1}_4+3A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 5V6ψ+6V5ψ+10V4ψ+14V3ψ+15V2ψ+10Vψ+6V05V_{6\psi}+6V_{5\psi}+10V_{4\psi}+14V_{3\psi}+15V_{2\psi}+10V_{\psi}+6V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=6h8+12h7+18h6+24h5+30h4+20h3+15h2+10h1e=3g84+3g56+2g54+g53+4g52+2g51f=g51+g52+g53+g54+g56+g84\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+15h_{2}+10h_{1}\\ e&=&3g_{84}+3g_{56}+2g_{54}+g_{53}+4g_{52}+2g_{51}\\ f&=&g_{-51}+g_{-52}+g_{-53}+g_{-54}+g_{-56}+g_{-84}\end{array}
Lie brackets of the above elements.
h=6h8+12h7+18h6+24h5+30h4+20h3+15h2+10h1e=3g84+3g56+2g54+g53+4g52+2g51f=g51+g52+g53+g54+g56+g84\begin{array}{rcl}[e, f]&=&6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+15h_{2}+10h_{1}\\ [h, e]&=&6g_{84}+6g_{56}+4g_{54}+2g_{53}+8g_{52}+4g_{51}\\ [h, f]&=&-2g_{-51}-2g_{-52}-2g_{-53}-2g_{-54}-2g_{-56}-2g_{-84}\end{array}
Centralizer type: A124+A1A^{24}_1+A_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 6): g2g_{-2}, h2h_{2}, h7+3h6+h5h3h1h_{7}+3h_{6}+h_{5}-h_{3}-h_{1}, g2g_{2}, g8+13g7+23g513g3+13g113g29g_{8}+1/3g_{7}+2/3g_{5}-1/3g_{3}+1/3g_{1}-1/3g_{-29}, g2913g1+13g313g523g713g8g_{29}-1/3g_{-1}+1/3g_{-3}-1/3g_{-5}-2/3g_{-7}-1/3g_{-8}
Basis of centralizer intersected with cartan (dimension: 2): h7+3h6+h5h3h1h_{7}+3h_{6}+h_{5}-h_{3}-h_{1}, h2-h_{2}
Cartan of centralizer (dimension: 2): h7+3h6+h5h3h1h_{7}+3h_{6}+h_{5}-h_{3}-h_{1}, h2-h_{2}
Cartan-generating semisimple element: h7+3h6+h5h3h2h1h_{7}+3h_{6}+h_{5}-h_{3}-h_{2}-h_{1}
adjoint action: (200000000000000000000200000010000001)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
Characteristic polynomial ad H: x65x4+4x2x^6-5x^4+4x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -2)(x -1)(x +1)(x +2)
Eigenvalues of ad H: 00, 22, 11, 1-1, 2-2
6 eigenvectors of ad H: 0, 1, 0, 0, 0, 0(0,1,0,0,0,0), 0, 0, 1, 0, 0, 0(0,0,1,0,0,0), 1, 0, 0, 0, 0, 0(1,0,0,0,0,0), 0, 0, 0, 0, 0, 1(0,0,0,0,0,1), 0, 0, 0, 0, 1, 0(0,0,0,0,1,0), 0, 0, 0, 1, 0, 0(0,0,0,1,0,0)
Centralizer type: A^{24}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed: (130)\begin{pmatrix}1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h2-h_{2}
matching e: g2g_{-2}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000200000000000000)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h2-h_{2}
matching e: g2g_{-2}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000200000000000000)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120)\begin{pmatrix}120\\ \end{pmatrix}

Scalar product computed: (1720)\begin{pmatrix}1/720\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
2h7+6h6+2h52h32h12h_{7}+6h_{6}+2h_{5}-2h_{3}-2h_{1}
matching e: g2913g1+13g313g523g713g8g_{29}-1/3g_{-1}+1/3g_{-3}-1/3g_{-5}-2/3g_{-7}-1/3g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000000020000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
2h7+6h6+2h52h32h12h_{7}+6h_{6}+2h_{5}-2h_{3}-2h_{1}
matching e: g2913g1+13g313g523g713g8g_{29}-1/3g_{-1}+1/3g_{-3}-1/3g_{-5}-2/3g_{-7}-1/3g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000000020000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (2880)\begin{pmatrix}2880\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=6h8+12h7+18h6+24h5+30h4+20h3+15h2+10h1e=x1g84+x3g56+x5g54+x6g53+x2g52+x4g51e=x10g51+x8g52+x12g53+x11g54+x9g56+x7g84\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+15h_{2}+10h_{1}\\ e&=&x_{1} g_{84}+x_{3} g_{56}+x_{5} g_{54}+x_{6} g_{53}+x_{2} g_{52}+x_{4} g_{51}\\ f&=&x_{10} g_{-51}+x_{8} g_{-52}+x_{12} g_{-53}+x_{11} g_{-54}+x_{9} g_{-56}+x_{7} g_{-84}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x7+x3x96)h8+(2x1x7+x3x9+x5x11+x6x1212)h7+(2x1x7+x2x8+x3x9+x4x10+x5x11+x6x1218)h6+(2x1x7+2x2x8+x3x9+x4x10+2x5x11+x6x1224)h5+(2x1x7+2x2x8+2x3x9+2x4x10+2x5x11+2x6x1230)h4+(2x1x7+x2x8+x3x9+2x4x10+x5x11+x6x1220)h3+(x1x7+x2x8+x3x9+x4x10+x5x11+x6x1215)h2+(x1x7+x2x8+x4x10+x6x1210)h1[e,f] - h = \left(x_{1} x_{7} +x_{3} x_{9} -6\right)h_{8}+\left(2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -12\right)h_{7}+\left(2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -18\right)h_{6}+\left(2x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -24\right)h_{5}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -30\right)h_{4}+\left(2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +2x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -20\right)h_{3}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -15\right)h_{2}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{6} x_{12} -10\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x7+x2x8+x4x10+x6x1210=0x1x7+x2x8+x3x9+x4x10+x5x11+x6x1215=02x1x7+x2x8+x3x9+2x4x10+x5x11+x6x1220=02x1x7+2x2x8+2x3x9+2x4x10+2x5x11+2x6x1230=02x1x7+2x2x8+x3x9+x4x10+2x5x11+x6x1224=02x1x7+x2x8+x3x9+x4x10+x5x11+x6x1218=02x1x7+x3x9+x5x11+x6x1212=0x1x7+x3x96=0\begin{array}{rcl}x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{6} x_{12} -10&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -15&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +2x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -30&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -24&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -18&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -12&=&0\\x_{1} x_{7} +x_{3} x_{9} -6&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=6h8+12h7+18h6+24h5+30h4+20h3+15h2+10h1e=x1g84+x3g56+x5g54+x6g53+x2g52+x4g51f=g51+g52+g53+g54+g56+g84\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+15h_{2}+10h_{1}\\e&=&x_{1} g_{84}+x_{3} g_{56}+x_{5} g_{54}+x_{6} g_{53}+x_{2} g_{52}+x_{4} g_{51}\\f&=&g_{-51}+g_{-52}+g_{-53}+g_{-54}+g_{-56}+g_{-84}\end{array}
Matrix form of the system we are trying to solve: (110101111111211211222222221121211111201011101000)[col. vect.]=(101520302418126)\begin{pmatrix}1 & 1 & 0 & 1 & 0 & 1\\ 1 & 1 & 1 & 1 & 1 & 1\\ 2 & 1 & 1 & 2 & 1 & 1\\ 2 & 2 & 2 & 2 & 2 & 2\\ 2 & 2 & 1 & 1 & 2 & 1\\ 2 & 1 & 1 & 1 & 1 & 1\\ 2 & 0 & 1 & 0 & 1 & 1\\ 1 & 0 & 1 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 15\\ 20\\ 30\\ 24\\ 18\\ 12\\ 6\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=6h8+12h7+18h6+24h5+30h4+20h3+15h2+10h1e=x1g84+x3g56+x5g54+x6g53+x2g52+x4g51f=x10g51+x8g52+x12g53+x11g54+x9g56+x7g84\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+24h_{5}+30h_{4}+20h_{3}+15h_{2}+10h_{1}\\ e&=&x_{1} g_{84}+x_{3} g_{56}+x_{5} g_{54}+x_{6} g_{53}+x_{2} g_{52}+x_{4} g_{51}\\ f&=&x_{10} g_{-51}+x_{8} g_{-52}+x_{12} g_{-53}+x_{11} g_{-54}+x_{9} g_{-56}+x_{7} g_{-84}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x7+x2x8+x4x10+x6x1210=0x1x7+x2x8+x3x9+x4x10+x5x11+x6x1215=02x1x7+x2x8+x3x9+2x4x10+x5x11+x6x1220=02x1x7+2x2x8+2x3x9+2x4x10+2x5x11+2x6x1230=02x1x7+2x2x8+x3x9+x4x10+2x5x11+x6x1224=02x1x7+x2x8+x3x9+x4x10+x5x11+x6x1218=02x1x7+x3x9+x5x11+x6x1212=0x1x7+x3x96=0\begin{array}{rcl}x_{1} x_{7} +x_{2} x_{8} +x_{4} x_{10} +x_{6} x_{12} -10&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -15&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +2x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -30&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -24&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -18&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -12&=&0\\x_{1} x_{7} +x_{3} x_{9} -6&=&0\\\end{array}

A114A^{14}_1
h-characteristic: (1, 0, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 28
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: A3+4A1A^{1}_3+4A^{1}_1 Containing regular semisimple subalgebra number 2: A3+A2A^{1}_3+A^{1}_2 Containing regular semisimple subalgebra number 3: D4+2A1D^{1}_4+2A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 3V6ψ+8V5ψ+8V4ψ+16V3ψ+16V2ψ+8Vψ+11V03V_{6\psi}+8V_{5\psi}+8V_{4\psi}+16V_{3\psi}+16V_{2\psi}+8V_{\psi}+11V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=6h8+12h7+18h6+23h5+28h4+19h3+14h2+10h1e=3g82+g70+4g60+g57+3g49+g47+g45f=g45+g47+g49+g57+g60+g70+g82\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+23h_{5}+28h_{4}+19h_{3}+14h_{2}+10h_{1}\\ e&=&3g_{82}+g_{70}+4g_{60}+g_{57}+3g_{49}+g_{47}+g_{45}\\ f&=&g_{-45}+g_{-47}+g_{-49}+g_{-57}+g_{-60}+g_{-70}+g_{-82}\end{array}
Lie brackets of the above elements.
h=6h8+12h7+18h6+23h5+28h4+19h3+14h2+10h1e=3g82+g70+4g60+g57+3g49+g47+g45f=g45+g47+g49+g57+g60+g70+g82\begin{array}{rcl}[e, f]&=&6h_{8}+12h_{7}+18h_{6}+23h_{5}+28h_{4}+19h_{3}+14h_{2}+10h_{1}\\ [h, e]&=&6g_{82}+2g_{70}+8g_{60}+2g_{57}+6g_{49}+2g_{47}+2g_{45}\\ [h, f]&=&-2g_{-45}-2g_{-47}-2g_{-49}-2g_{-57}-2g_{-60}-2g_{-70}-2g_{-82}\end{array}
Centralizer type: B2B_2
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 11): h5h3h_{5}-h_{3}, g3g5g_{3}-g_{-5}, g5g3g_{5}-g_{-3}, g7+g4+g4+g7g_{7}+g_{4}+g_{-4}+g_{-7}, g11g12g_{11}-g_{-12}, g12g11g_{12}-g_{-11}, g17+g18g_{17}+g_{-18}, g18+g17g_{18}+g_{-17}, g19+g4+g4+g19g_{19}+g_{4}+g_{-4}+g_{-19}, g25+g10g10g25g_{25}+g_{10}-g_{-10}-g_{-25}, g31g2g2+g31g_{31}-g_{2}-g_{-2}+g_{-31}
Basis of centralizer intersected with cartan (dimension: 1): h5h3h_{5}-h_{3}
Cartan of centralizer (dimension: 3): 8g31+6g25+8g19+3g18+7g17+4g12+4g11+6g10+8g4+4g38g2+h5h38g2+8g44g56g104g114g12+3g17+7g18+8g196g25+8g318g_{31}+6g_{25}+8g_{19}+3g_{18}+7g_{17}+4g_{12}+4g_{11}+6g_{10}+8g_{4}+4g_{3}-8g_{2}+h_{5}-h_{3}-8g_{-2}+8g_{-4}-4g_{-5}-6g_{-10}-4g_{-11}-4g_{-12}+3g_{-17}+7g_{-18}+8g_{-19}-6g_{-25}+8g_{-31}, 7g315g25132g192g186g173g123g115g10+g5132g43g3+7g2+7g2g3132g4+3g5+5g10+3g11+3g122g176g18132g19+5g257g31-7g_{31}-5g_{25}-13/2g_{19}-2g_{18}-6g_{17}-3g_{12}-3g_{11}-5g_{10}+g_{5}-13/2g_{4}-3g_{3}+7g_{2}+7g_{-2}-g_{-3}-13/2g_{-4}+3g_{-5}+5g_{-10}+3g_{-11}+3g_{-12}-2g_{-17}-6g_{-18}-13/2g_{-19}+5g_{-25}-7g_{-31}, 12g19g712g412g4g7+12g191/2g_{19}-g_{7}-1/2g_{4}-1/2g_{-4}-g_{-7}+1/2g_{-19}
Cartan-generating semisimple element: g31+g25+g19+g18+g17+g12+g11+g10+g7+g5+2g4+g3g2+h5h3g2g3+2g4g5+g7g10g11g12+g17+g18+g19g25+g31g_{31}+g_{25}+g_{19}+g_{18}+g_{17}+g_{12}+g_{11}+g_{10}+g_{7}+g_{5}+2g_{4}+g_{3}-g_{2}+h_{5}-h_{3}-g_{-2}-g_{-3}+2g_{-4}-g_{-5}+g_{-7}-g_{-10}-g_{-11}-g_{-12}+g_{-17}+g_{-18}+g_{-19}-g_{-25}+g_{-31}
adjoint action: (0110111100022013020220202103022200000000000023012020202203102022022200202002220200202022011011000220111001120300011111230)\begin{pmatrix}0 & -1 & 1 & 0 & -1 & 1 & 1 & -1 & 0 & 0 & 0\\ 2 & -2 & 0 & 1 & -3 & 0 & 2 & 0 & 2 & -2 & 0\\ -2 & 0 & 2 & -1 & 0 & 3 & 0 & -2 & -2 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2 & -3 & 0 & 1 & -2 & 0 & -2 & 0 & 2 & 0 & 2\\ -2 & 0 & 3 & -1 & 0 & 2 & 0 & -2 & -2 & 0 & 2\\ 2 & -2 & 0 & 0 & -2 & 0 & -2 & 0 & 0 & 2 & 2\\ -2 & 0 & 2 & 0 & 0 & -2 & 0 & 2 & 0 & -2 & 2\\ 0 & -1 & 1 & 0 & -1 & 1 & 0 & 0 & 0 & -2 & 2\\ 0 & -1 & 1 & -1 & 0 & 0 & -1 & 1 & -2 & 0 & 3\\ 0 & 0 & 0 & -1 & 1 & 1 & -1 & -1 & -2 & 3 & 0\\ \end{pmatrix}
Characteristic polynomial ad H: x1127x9+219x7585x5+392x3x^{11}-27x^9+219x^7-585x^5+392x^3
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -1)(x +1)(x^2-8)(x^2-2x -7)(x^2+2x -7)
Eigenvalues of ad H: 00, 11, 1-1, 222\sqrt{2}, 22-2\sqrt{2}, 22+12\sqrt{2}+1, 22+1-2\sqrt{2}+1, 2212\sqrt{2}-1, 221-2\sqrt{2}-1
11 eigenvectors of ad H: 0, 0, 0, -2, 0, 0, 0, 0, 1, 0, 0(0,0,0,2,0,0,0,0,1,0,0), 7/2, 2, 4, 4, 2, 2, 1/2, 5/2, 0, 1, 0(72,2,4,4,2,2,12,52,0,1,0), -5/2, -1, -3, -1, -1, -1, 1/2, -3/2, 0, 0, 1(52,1,3,1,1,1,12,32,0,0,1), -2, -1, -5, 0, -1, -3, 0, -4, -2, -1, 1(2,1,5,0,1,3,0,4,2,1,1), -14, -15, -11, 0, -1, -3, 8, -4, -2, 1, 1(14,15,11,0,1,3,8,4,2,1,1), -1/4\sqrt{2}-1, -1, -\sqrt{2}-1, 0, -1, -3, 3/4\sqrt{2}+1/2, 7/4\sqrt{2}-1/2, -2, 3/2\sqrt{2}, 1(1421,1,21,0,1,3,342+12,74212,2,322,1), 1/4\sqrt{2}-1, -1, \sqrt{2}-1, 0, -1, -3, -3/4\sqrt{2}+1/2, -7/4\sqrt{2}-1/2, -2, -3/2\sqrt{2}, 1(1421,1,21,0,1,3,342+12,74212,2,322,1), 2\sqrt{2}+2, \sqrt{2}+1, 3\sqrt{2}+5, 0, \sqrt{2}+1, 5\sqrt{2}+3, 0, -4, 2\sqrt{2}+2, -1, 1(22+2,2+1,32+5,0,2+1,52+3,0,4,22+2,1,1), -2\sqrt{2}+2, -\sqrt{2}+1, -3\sqrt{2}+5, 0, -\sqrt{2}+1, -5\sqrt{2}+3, 0, -4, -2\sqrt{2}+2, -1, 1(22+2,2+1,32+5,0,2+1,52+3,0,4,22+2,1,1), 6/31\sqrt{2}-14/31, 33/31\sqrt{2}-15/31, -13/31\sqrt{2}-11/31, 0, 15/31\sqrt{2}-97/31, 11/31\sqrt{2}-67/31, 60/31\sqrt{2}-16/31, 44/31\sqrt{2}-20/31, 14/31\sqrt{2}-74/31, 56/31\sqrt{2}-17/31, 1(63121431,333121531,133121131,0,153129731,113126731,603121631,443122031,143127431,563121731,1), -6/31\sqrt{2}-14/31, -33/31\sqrt{2}-15/31, 13/31\sqrt{2}-11/31, 0, -15/31\sqrt{2}-97/31, -11/31\sqrt{2}-67/31, -60/31\sqrt{2}-16/31, -44/31\sqrt{2}-20/31, -14/31\sqrt{2}-74/31, -56/31\sqrt{2}-17/31, 1(63121431,333121531,133121131,0,153129731,113126731,603121631,443122031,143127431,563121731,1)
Centralizer type: B^{1}_2
Reductive components (1 total):
Scalar product computed: (160160160130)\begin{pmatrix}1/60 & -1/60\\ -1/60 & 1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
27g31+27g25+57g19+87g18+67g12+67g11+27g10+2g5+57g4+67g3+27g2+127h5127h3+27g22g3+57g467g527g1067g1167g12+87g17+57g1927g2527g31-2/7g_{31}+2/7g_{25}+5/7g_{19}+8/7g_{18}+6/7g_{12}+6/7g_{11}+2/7g_{10}+2g_{5}+5/7g_{4}+6/7g_{3}+2/7g_{2}+12/7h_{5}-12/7h_{3}+2/7g_{-2}-2g_{-3}+5/7g_{-4}-6/7g_{-5}-2/7g_{-10}-6/7g_{-11}-6/7g_{-12}+8/7g_{-17}+5/7g_{-19}-2/7g_{-25}-2/7g_{-31}
matching e: g31g252g194g183g12g11g105g52g4g3g22h5+2h3g2+5g32g4+g5+g10+3g11+g124g172g19+g25+g31g_{31}-g_{25}-2g_{19}-4g_{18}-3g_{12}-g_{11}-g_{10}-5g_{5}-2g_{4}-g_{3}-g_{2}-2h_{5}+2h_{3}-g_{-2}+5g_{-3}-2g_{-4}+g_{-5}+g_{-10}+3g_{-11}+g_{-12}-4g_{-17}-2g_{-19}+g_{-25}+g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (026706767870000127247067107047012700402476701070471271670000000000001271070672470470127001270107202470474016704700470247001271271670470047024704127067670267000474708702700267470107000278706767471070)\begin{pmatrix}0 & -2 & 6/7 & 0 & -6/7 & 6/7 & 8/7 & 0 & 0 & 0 & 0\\ 12/7 & -24/7 & 0 & 6/7 & -10/7 & 0 & 4/7 & 0 & 12/7 & 0 & 0\\ -4 & 0 & 24/7 & -6/7 & 0 & 10/7 & 0 & -4/7 & -12/7 & 16/7 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 12/7 & -10/7 & 0 & 6/7 & -24/7 & 0 & 4/7 & 0 & 12/7 & 0 & 0\\ -12/7 & 0 & 10/7 & -2 & 0 & 24/7 & 0 & 4/7 & -4 & 0 & 16/7\\ 0 & -4/7 & 0 & 0 & 4/7 & 0 & -24/7 & 0 & 0 & 12/7 & 12/7\\ -16/7 & 0 & 4/7 & 0 & 0 & 4/7 & 0 & 24/7 & 0 & -4 & 12/7\\ 0 & -6/7 & 6/7 & 0 & -2 & 6/7 & 0 & 0 & 0 & 4/7 & 4/7\\ 0 & -8/7 & 0 & 2/7 & 0 & 0 & -2 & 6/7 & 4/7 & 0 & 10/7\\ 0 & 0 & 0 & -2/7 & 8/7 & 0 & -6/7 & -6/7 & -4/7 & 10/7 & 0\\ \end{pmatrix}
(272+17)g31+(314217)g25+(272514)g19+(328247)g18+142g17+(17237)g12+(17237)g11+(314217)g10g5+(272514)g4+(17237)g3+(27217)g2+(128267)h5+(1282+67)h3+(27217)g2+g3+(272514)g4+(172+37)g5+(3142+17)g10+(172+37)g11+(172+37)g12+(328247)g17+142g18+(272514)g19+(3142+17)g25+(272+17)g31\left(2/7\sqrt{2}+1/7\right)g_{31}+\left(3/14\sqrt{2}-1/7\right)g_{25}+\left(2/7\sqrt{2}-5/14\right)g_{19}+\left(3/28\sqrt{2}-4/7\right)g_{18}+1/4\sqrt{2}g_{17}+\left(1/7\sqrt{2}-3/7\right)g_{12}+\left(1/7\sqrt{2}-3/7\right)g_{11}+\left(3/14\sqrt{2}-1/7\right)g_{10}-g_{5}+\left(2/7\sqrt{2}-5/14\right)g_{4}+\left(1/7\sqrt{2}-3/7\right)g_{3}+\left(-2/7\sqrt{2}-1/7\right)g_{2}+\left(1/28\sqrt{2}-6/7\right)h_{5}+\left(-1/28\sqrt{2}+6/7\right)h_{3}+\left(-2/7\sqrt{2}-1/7\right)g_{-2}+g_{-3}+\left(2/7\sqrt{2}-5/14\right)g_{-4}+\left(-1/7\sqrt{2}+3/7\right)g_{-5}+\left(-3/14\sqrt{2}+1/7\right)g_{-10}+\left(-1/7\sqrt{2}+3/7\right)g_{-11}+\left(-1/7\sqrt{2}+3/7\right)g_{-12}+\left(3/28\sqrt{2}-4/7\right)g_{-17}+1/4\sqrt{2}g_{-18}+\left(2/7\sqrt{2}-5/14\right)g_{-19}+\left(-3/14\sqrt{2}+1/7\right)g_{-25}+\left(2/7\sqrt{2}+1/7\right)g_{-31}
matching e: g31+(563121731)g25+(143127431)g19+(443122031)g18+(603121631)g17+(113126731)g12+(153129731)g11+(563121731)g10+(133121131)g5+(143127431)g4+(333121531)g3g2+(63121431)h5+(6312+1431)h3g2+(13312+1131)g3+(143127431)g4+(33312+1531)g5+(56312+1731)g10+(11312+6731)g11+(15312+9731)g12+(443122031)g17+(603121631)g18+(143127431)g19+(56312+1731)g25+g31g_{31}+\left(56/31\sqrt{2}-17/31\right)g_{25}+\left(14/31\sqrt{2}-74/31\right)g_{19}+\left(44/31\sqrt{2}-20/31\right)g_{18}+\left(60/31\sqrt{2}-16/31\right)g_{17}+\left(11/31\sqrt{2}-67/31\right)g_{12}+\left(15/31\sqrt{2}-97/31\right)g_{11}+\left(56/31\sqrt{2}-17/31\right)g_{10}+\left(-13/31\sqrt{2}-11/31\right)g_{5}+\left(14/31\sqrt{2}-74/31\right)g_{4}+\left(33/31\sqrt{2}-15/31\right)g_{3}-g_{2}+\left(6/31\sqrt{2}-14/31\right)h_{5}+\left(-6/31\sqrt{2}+14/31\right)h_{3}-g_{-2}+\left(13/31\sqrt{2}+11/31\right)g_{-3}+\left(14/31\sqrt{2}-74/31\right)g_{-4}+\left(-33/31\sqrt{2}+15/31\right)g_{-5}+\left(-56/31\sqrt{2}+17/31\right)g_{-10}+\left(-11/31\sqrt{2}+67/31\right)g_{-11}+\left(-15/31\sqrt{2}+97/31\right)g_{-12}+\left(44/31\sqrt{2}-20/31\right)g_{-17}+\left(60/31\sqrt{2}-16/31\right)g_{-18}+\left(14/31\sqrt{2}-74/31\right)g_{-19}+\left(-56/31\sqrt{2}+17/31\right)g_{-25}+g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (01172370172+3717237328247142000272671142+127017237472+570372270272671220201142127172+370472570372+27272+6731428700000000000027267472+570172371142+1270472270272670122272+6704725710114212704722720314287122372+27004722701142+1270027267272673142+8703722700472270114212702272670172+37172370117237000472273722703282+471422721700117237472270472570003142+17328247142172+37172+37372+27472570)\begin{pmatrix}0 & 1 & 1/7\sqrt{2}-3/7 & 0 & -1/7\sqrt{2}+3/7 & 1/7\sqrt{2}-3/7 & 3/28\sqrt{2}-4/7 & -1/4\sqrt{2} & 0 & 0 & 0\\ 2/7\sqrt{2}-6/7 & -1/14\sqrt{2}+12/7 & 0 & 1/7\sqrt{2}-3/7 & -4/7\sqrt{2}+5/7 & 0 & 3/7\sqrt{2}-2/7 & 0 & 2/7\sqrt{2}-6/7 & -1/2\sqrt{2} & 0\\ 2 & 0 & 1/14\sqrt{2}-12/7 & -1/7\sqrt{2}+3/7 & 0 & 4/7\sqrt{2}-5/7 & 0 & -3/7\sqrt{2}+2/7 & -2/7\sqrt{2}+6/7 & 3/14\sqrt{2}-8/7 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/7\sqrt{2}-6/7 & -4/7\sqrt{2}+5/7 & 0 & 1/7\sqrt{2}-3/7 & -1/14\sqrt{2}+12/7 & 0 & -4/7\sqrt{2}-2/7 & 0 & 2/7\sqrt{2}-6/7 & 0 & 1/2\sqrt{2}\\ -2/7\sqrt{2}+6/7 & 0 & 4/7\sqrt{2}-5/7 & 1 & 0 & 1/14\sqrt{2}-12/7 & 0 & -4/7\sqrt{2}-2/7 & 2 & 0 & 3/14\sqrt{2}-8/7\\ 1/2\sqrt{2} & -3/7\sqrt{2}+2/7 & 0 & 0 & -4/7\sqrt{2}-2/7 & 0 & -1/14\sqrt{2}+12/7 & 0 & 0 & 2/7\sqrt{2}-6/7 & 2/7\sqrt{2}-6/7\\ -3/14\sqrt{2}+8/7 & 0 & 3/7\sqrt{2}-2/7 & 0 & 0 & -4/7\sqrt{2}-2/7 & 0 & 1/14\sqrt{2}-12/7 & 0 & 2 & 2/7\sqrt{2}-6/7\\ 0 & -1/7\sqrt{2}+3/7 & 1/7\sqrt{2}-3/7 & 0 & 1 & 1/7\sqrt{2}-3/7 & 0 & 0 & 0 & -4/7\sqrt{2}-2/7 & 3/7\sqrt{2}-2/7\\ 0 & -3/28\sqrt{2}+4/7 & 1/4\sqrt{2} & -2/7\sqrt{2}-1/7 & 0 & 0 & 1 & 1/7\sqrt{2}-3/7 & -4/7\sqrt{2}-2/7 & 0 & 4/7\sqrt{2}-5/7\\ 0 & 0 & 0 & -3/14\sqrt{2}+1/7 & 3/28\sqrt{2}-4/7 & 1/4\sqrt{2} & -1/7\sqrt{2}+3/7 & -1/7\sqrt{2}+3/7 & -3/7\sqrt{2}+2/7 & 4/7\sqrt{2}-5/7 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
27g31+27g25+57g19+87g18+67g12+67g11+27g10+2g5+57g4+67g3+27g2+127h5127h3+27g22g3+57g467g527g1067g1167g12+87g17+57g1927g2527g31-2/7g_{31}+2/7g_{25}+5/7g_{19}+8/7g_{18}+6/7g_{12}+6/7g_{11}+2/7g_{10}+2g_{5}+5/7g_{4}+6/7g_{3}+2/7g_{2}+12/7h_{5}-12/7h_{3}+2/7g_{-2}-2g_{-3}+5/7g_{-4}-6/7g_{-5}-2/7g_{-10}-6/7g_{-11}-6/7g_{-12}+8/7g_{-17}+5/7g_{-19}-2/7g_{-25}-2/7g_{-31}
matching e: g31g252g194g183g12g11g105g52g4g3g22h5+2h3g2+5g32g4+g5+g10+3g11+g124g172g19+g25+g31g_{31}-g_{25}-2g_{19}-4g_{18}-3g_{12}-g_{11}-g_{10}-5g_{5}-2g_{4}-g_{3}-g_{2}-2h_{5}+2h_{3}-g_{-2}+5g_{-3}-2g_{-4}+g_{-5}+g_{-10}+3g_{-11}+g_{-12}-4g_{-17}-2g_{-19}+g_{-25}+g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (026706767870000127247067107047012700402476701070471271670000000000001271070672470470127001270107202470474016704700470247001271271670470047024704127067670267000474708702700267470107000278706767471070)\begin{pmatrix}0 & -2 & 6/7 & 0 & -6/7 & 6/7 & 8/7 & 0 & 0 & 0 & 0\\ 12/7 & -24/7 & 0 & 6/7 & -10/7 & 0 & 4/7 & 0 & 12/7 & 0 & 0\\ -4 & 0 & 24/7 & -6/7 & 0 & 10/7 & 0 & -4/7 & -12/7 & 16/7 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 12/7 & -10/7 & 0 & 6/7 & -24/7 & 0 & 4/7 & 0 & 12/7 & 0 & 0\\ -12/7 & 0 & 10/7 & -2 & 0 & 24/7 & 0 & 4/7 & -4 & 0 & 16/7\\ 0 & -4/7 & 0 & 0 & 4/7 & 0 & -24/7 & 0 & 0 & 12/7 & 12/7\\ -16/7 & 0 & 4/7 & 0 & 0 & 4/7 & 0 & 24/7 & 0 & -4 & 12/7\\ 0 & -6/7 & 6/7 & 0 & -2 & 6/7 & 0 & 0 & 0 & 4/7 & 4/7\\ 0 & -8/7 & 0 & 2/7 & 0 & 0 & -2 & 6/7 & 4/7 & 0 & 10/7\\ 0 & 0 & 0 & -2/7 & 8/7 & 0 & -6/7 & -6/7 & -4/7 & 10/7 & 0\\ \end{pmatrix}
(272+17)g31+(314217)g25+(272514)g19+(328247)g18+142g17+(17237)g12+(17237)g11+(314217)g10g5+(272514)g4+(17237)g3+(27217)g2+(128267)h5+(1282+67)h3+(27217)g2+g3+(272514)g4+(172+37)g5+(3142+17)g10+(172+37)g11+(172+37)g12+(328247)g17+142g18+(272514)g19+(3142+17)g25+(272+17)g31\left(2/7\sqrt{2}+1/7\right)g_{31}+\left(3/14\sqrt{2}-1/7\right)g_{25}+\left(2/7\sqrt{2}-5/14\right)g_{19}+\left(3/28\sqrt{2}-4/7\right)g_{18}+1/4\sqrt{2}g_{17}+\left(1/7\sqrt{2}-3/7\right)g_{12}+\left(1/7\sqrt{2}-3/7\right)g_{11}+\left(3/14\sqrt{2}-1/7\right)g_{10}-g_{5}+\left(2/7\sqrt{2}-5/14\right)g_{4}+\left(1/7\sqrt{2}-3/7\right)g_{3}+\left(-2/7\sqrt{2}-1/7\right)g_{2}+\left(1/28\sqrt{2}-6/7\right)h_{5}+\left(-1/28\sqrt{2}+6/7\right)h_{3}+\left(-2/7\sqrt{2}-1/7\right)g_{-2}+g_{-3}+\left(2/7\sqrt{2}-5/14\right)g_{-4}+\left(-1/7\sqrt{2}+3/7\right)g_{-5}+\left(-3/14\sqrt{2}+1/7\right)g_{-10}+\left(-1/7\sqrt{2}+3/7\right)g_{-11}+\left(-1/7\sqrt{2}+3/7\right)g_{-12}+\left(3/28\sqrt{2}-4/7\right)g_{-17}+1/4\sqrt{2}g_{-18}+\left(2/7\sqrt{2}-5/14\right)g_{-19}+\left(-3/14\sqrt{2}+1/7\right)g_{-25}+\left(2/7\sqrt{2}+1/7\right)g_{-31}
matching e: g31+(563121731)g25+(143127431)g19+(443122031)g18+(603121631)g17+(113126731)g12+(153129731)g11+(563121731)g10+(133121131)g5+(143127431)g4+(333121531)g3g2+(63121431)h5+(6312+1431)h3g2+(13312+1131)g3+(143127431)g4+(33312+1531)g5+(56312+1731)g10+(11312+6731)g11+(15312+9731)g12+(443122031)g17+(603121631)g18+(143127431)g19+(56312+1731)g25+g31g_{31}+\left(56/31\sqrt{2}-17/31\right)g_{25}+\left(14/31\sqrt{2}-74/31\right)g_{19}+\left(44/31\sqrt{2}-20/31\right)g_{18}+\left(60/31\sqrt{2}-16/31\right)g_{17}+\left(11/31\sqrt{2}-67/31\right)g_{12}+\left(15/31\sqrt{2}-97/31\right)g_{11}+\left(56/31\sqrt{2}-17/31\right)g_{10}+\left(-13/31\sqrt{2}-11/31\right)g_{5}+\left(14/31\sqrt{2}-74/31\right)g_{4}+\left(33/31\sqrt{2}-15/31\right)g_{3}-g_{2}+\left(6/31\sqrt{2}-14/31\right)h_{5}+\left(-6/31\sqrt{2}+14/31\right)h_{3}-g_{-2}+\left(13/31\sqrt{2}+11/31\right)g_{-3}+\left(14/31\sqrt{2}-74/31\right)g_{-4}+\left(-33/31\sqrt{2}+15/31\right)g_{-5}+\left(-56/31\sqrt{2}+17/31\right)g_{-10}+\left(-11/31\sqrt{2}+67/31\right)g_{-11}+\left(-15/31\sqrt{2}+97/31\right)g_{-12}+\left(44/31\sqrt{2}-20/31\right)g_{-17}+\left(60/31\sqrt{2}-16/31\right)g_{-18}+\left(14/31\sqrt{2}-74/31\right)g_{-19}+\left(-56/31\sqrt{2}+17/31\right)g_{-25}+g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (01172370172+3717237328247142000272671142+127017237472+570372270272671220201142127172+370472570372+27272+6731428700000000000027267472+570172371142+1270472270272670122272+6704725710114212704722720314287122372+27004722701142+1270027267272673142+8703722700472270114212702272670172+37172370117237000472273722703282+471422721700117237472270472570003142+17328247142172+37172+37372+27472570)\begin{pmatrix}0 & 1 & 1/7\sqrt{2}-3/7 & 0 & -1/7\sqrt{2}+3/7 & 1/7\sqrt{2}-3/7 & 3/28\sqrt{2}-4/7 & -1/4\sqrt{2} & 0 & 0 & 0\\ 2/7\sqrt{2}-6/7 & -1/14\sqrt{2}+12/7 & 0 & 1/7\sqrt{2}-3/7 & -4/7\sqrt{2}+5/7 & 0 & 3/7\sqrt{2}-2/7 & 0 & 2/7\sqrt{2}-6/7 & -1/2\sqrt{2} & 0\\ 2 & 0 & 1/14\sqrt{2}-12/7 & -1/7\sqrt{2}+3/7 & 0 & 4/7\sqrt{2}-5/7 & 0 & -3/7\sqrt{2}+2/7 & -2/7\sqrt{2}+6/7 & 3/14\sqrt{2}-8/7 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/7\sqrt{2}-6/7 & -4/7\sqrt{2}+5/7 & 0 & 1/7\sqrt{2}-3/7 & -1/14\sqrt{2}+12/7 & 0 & -4/7\sqrt{2}-2/7 & 0 & 2/7\sqrt{2}-6/7 & 0 & 1/2\sqrt{2}\\ -2/7\sqrt{2}+6/7 & 0 & 4/7\sqrt{2}-5/7 & 1 & 0 & 1/14\sqrt{2}-12/7 & 0 & -4/7\sqrt{2}-2/7 & 2 & 0 & 3/14\sqrt{2}-8/7\\ 1/2\sqrt{2} & -3/7\sqrt{2}+2/7 & 0 & 0 & -4/7\sqrt{2}-2/7 & 0 & -1/14\sqrt{2}+12/7 & 0 & 0 & 2/7\sqrt{2}-6/7 & 2/7\sqrt{2}-6/7\\ -3/14\sqrt{2}+8/7 & 0 & 3/7\sqrt{2}-2/7 & 0 & 0 & -4/7\sqrt{2}-2/7 & 0 & 1/14\sqrt{2}-12/7 & 0 & 2 & 2/7\sqrt{2}-6/7\\ 0 & -1/7\sqrt{2}+3/7 & 1/7\sqrt{2}-3/7 & 0 & 1 & 1/7\sqrt{2}-3/7 & 0 & 0 & 0 & -4/7\sqrt{2}-2/7 & 3/7\sqrt{2}-2/7\\ 0 & -3/28\sqrt{2}+4/7 & 1/4\sqrt{2} & -2/7\sqrt{2}-1/7 & 0 & 0 & 1 & 1/7\sqrt{2}-3/7 & -4/7\sqrt{2}-2/7 & 0 & 4/7\sqrt{2}-5/7\\ 0 & 0 & 0 & -3/14\sqrt{2}+1/7 & 3/28\sqrt{2}-4/7 & 1/4\sqrt{2} & -1/7\sqrt{2}+3/7 & -1/7\sqrt{2}+3/7 & -3/7\sqrt{2}+2/7 & 4/7\sqrt{2}-5/7 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1, 0), (-1, 0), (1, 2), (-1, -2), (1, 1), (-1, -1), (0, -1), (0, 1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240120120120)\begin{pmatrix}240 & -120\\ -120 & 120\\ \end{pmatrix}
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Unknown elements.
h=6h8+12h7+18h6+23h5+28h4+19h3+14h2+10h1e=x1g82+x4g70+x2g60+x6g57+x3g49+x5g47+x7g45e=x14g45+x12g47+x10g49+x13g57+x9g60+x11g70+x8g82\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+23h_{5}+28h_{4}+19h_{3}+14h_{2}+10h_{1}\\ e&=&x_{1} g_{82}+x_{4} g_{70}+x_{2} g_{60}+x_{6} g_{57}+x_{3} g_{49}+x_{5} g_{47}+x_{7} g_{45}\\ f&=&x_{14} g_{-45}+x_{12} g_{-47}+x_{10} g_{-49}+x_{13} g_{-57}+x_{9} g_{-60}+x_{11} g_{-70}+x_{8} g_{-82}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x8+x3x106)h8+(x1x8+x2x9+x3x10+x4x11+x5x1212)h7+(x1x8+2x2x9+x3x10+x4x11+x5x12+x6x13+x7x1418)h6+(2x1x8+2x2x9+x3x10+2x4x11+x5x12+2x6x13+x7x1423)h5+(3x1x8+2x2x9+x3x10+3x4x11+x5x12+2x6x13+2x7x1428)h4+(2x1x8+x2x9+x3x10+2x4x11+x5x12+2x6x13+x7x1419)h3+(2x1x8+x2x9+x4x11+x5x12+x6x13+x7x1414)h2+(x1x8+x3x10+x4x11+x5x12+x6x13+x7x1410)h1[e,f] - h = \left(x_{1} x_{8} +x_{3} x_{10} -6\right)h_{8}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} -12\right)h_{7}+\left(x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -18\right)h_{6}+\left(2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -23\right)h_{5}+\left(3x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +3x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -28\right)h_{4}+\left(2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -19\right)h_{3}+\left(2x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -14\right)h_{2}+\left(x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -10\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x3x10+x4x11+x5x12+x6x13+x7x1410=02x1x8+x2x9+x4x11+x5x12+x6x13+x7x1414=02x1x8+x2x9+x3x10+2x4x11+x5x12+2x6x13+x7x1419=03x1x8+2x2x9+x3x10+3x4x11+x5x12+2x6x13+2x7x1428=02x1x8+2x2x9+x3x10+2x4x11+x5x12+2x6x13+x7x1423=0x1x8+2x2x9+x3x10+x4x11+x5x12+x6x13+x7x1418=0x1x8+x2x9+x3x10+x4x11+x5x1212=0x1x8+x3x106=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -10&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -14&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -19&=&0\\3x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +3x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -28&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -23&=&0\\x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -18&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} -12&=&0\\x_{1} x_{8} +x_{3} x_{10} -6&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=6h8+12h7+18h6+23h5+28h4+19h3+14h2+10h1e=x1g82+x4g70+x2g60+x6g57+x3g49+x5g47+x7g45f=g45+g47+g49+g57+g60+g70+g82\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+23h_{5}+28h_{4}+19h_{3}+14h_{2}+10h_{1}\\e&=&x_{1} g_{82}+x_{4} g_{70}+x_{2} g_{60}+x_{6} g_{57}+x_{3} g_{49}+x_{5} g_{47}+x_{7} g_{45}\\f&=&g_{-45}+g_{-47}+g_{-49}+g_{-57}+g_{-60}+g_{-70}+g_{-82}\end{array}
Matrix form of the system we are trying to solve: (10111112101111211212132131222212121121111111111001010000)[col. vect.]=(101419282318126)\begin{pmatrix}1 & 0 & 1 & 1 & 1 & 1 & 1\\ 2 & 1 & 0 & 1 & 1 & 1 & 1\\ 2 & 1 & 1 & 2 & 1 & 2 & 1\\ 3 & 2 & 1 & 3 & 1 & 2 & 2\\ 2 & 2 & 1 & 2 & 1 & 2 & 1\\ 1 & 2 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 0 & 0\\ 1 & 0 & 1 & 0 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 14\\ 19\\ 28\\ 23\\ 18\\ 12\\ 6\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=6h8+12h7+18h6+23h5+28h4+19h3+14h2+10h1e=x1g82+x4g70+x2g60+x6g57+x3g49+x5g47+x7g45f=x14g45+x12g47+x10g49+x13g57+x9g60+x11g70+x8g82\begin{array}{rcl}h&=&6h_{8}+12h_{7}+18h_{6}+23h_{5}+28h_{4}+19h_{3}+14h_{2}+10h_{1}\\ e&=&x_{1} g_{82}+x_{4} g_{70}+x_{2} g_{60}+x_{6} g_{57}+x_{3} g_{49}+x_{5} g_{47}+x_{7} g_{45}\\ f&=&x_{14} g_{-45}+x_{12} g_{-47}+x_{10} g_{-49}+x_{13} g_{-57}+x_{9} g_{-60}+x_{11} g_{-70}+x_{8} g_{-82}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x3x10+x4x11+x5x12+x6x13+x7x1410=02x1x8+x2x9+x4x11+x5x12+x6x13+x7x1414=02x1x8+x2x9+x3x10+2x4x11+x5x12+2x6x13+x7x1419=03x1x8+2x2x9+x3x10+3x4x11+x5x12+2x6x13+2x7x1428=02x1x8+2x2x9+x3x10+2x4x11+x5x12+2x6x13+x7x1423=0x1x8+2x2x9+x3x10+x4x11+x5x12+x6x13+x7x1418=0x1x8+x2x9+x3x10+x4x11+x5x1212=0x1x8+x3x106=0\begin{array}{rcl}x_{1} x_{8} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -10&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -14&=&0\\2x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -19&=&0\\3x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +3x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -28&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +2x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -23&=&0\\x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -18&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} -12&=&0\\x_{1} x_{8} +x_{3} x_{10} -6&=&0\\\end{array}

A113A^{13}_1
h-characteristic: (0, 1, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 26
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: 3A2+A13A^{1}_2+A^{1}_1 Containing regular semisimple subalgebra number 2: A3+3A1A^{1}_3+3A^{1}_1 Containing regular semisimple subalgebra number 3: D4+A1D^{1}_4+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 2V6ψ+6V5ψ+13V4ψ+12V3ψ+16V2ψ+14Vψ+9V02V_{6\psi}+6V_{5\psi}+13V_{4\psi}+12V_{3\psi}+16V_{2\psi}+14V_{\psi}+9V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=6h8+12h7+17h6+22h5+27h4+18h3+14h2+9h1e=2g87+g69+2g65+2g58+2g56+2g55+2g54f=g54+g55+g56+g58+g65+g69+g87\begin{array}{rcl}h&=&6h_{8}+12h_{7}+17h_{6}+22h_{5}+27h_{4}+18h_{3}+14h_{2}+9h_{1}\\ e&=&2g_{87}+g_{69}+2g_{65}+2g_{58}+2g_{56}+2g_{55}+2g_{54}\\ f&=&g_{-54}+g_{-55}+g_{-56}+g_{-58}+g_{-65}+g_{-69}+g_{-87}\end{array}
Lie brackets of the above elements.
h=6h8+12h7+17h6+22h5+27h4+18h3+14h2+9h1e=2g87+g69+2g65+2g58+2g56+2g55+2g54f=g54+g55+g56+g58+g65+g69+g87\begin{array}{rcl}[e, f]&=&6h_{8}+12h_{7}+17h_{6}+22h_{5}+27h_{4}+18h_{3}+14h_{2}+9h_{1}\\ [h, e]&=&4g_{87}+2g_{69}+4g_{65}+4g_{58}+4g_{56}+4g_{55}+4g_{54}\\ [h, f]&=&-2g_{-54}-2g_{-55}-2g_{-56}-2g_{-58}-2g_{-65}-2g_{-69}-2g_{-87}\end{array}
Centralizer type: 3A13A_1
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Unknown elements.
h=6h8+12h7+17h6+22h5+27h4+18h3+14h2+9h1e=x1g87+x7g69+x3g65+x2g58+x4g56+x6g55+x5g54e=x12g54+x13g55+x11g56+x9g58+x10g65+x14g69+x8g87\begin{array}{rcl}h&=&6h_{8}+12h_{7}+17h_{6}+22h_{5}+27h_{4}+18h_{3}+14h_{2}+9h_{1}\\ e&=&x_{1} g_{87}+x_{7} g_{69}+x_{3} g_{65}+x_{2} g_{58}+x_{4} g_{56}+x_{6} g_{55}+x_{5} g_{54}\\ f&=&x_{12} g_{-54}+x_{13} g_{-55}+x_{11} g_{-56}+x_{9} g_{-58}+x_{10} g_{-65}+x_{14} g_{-69}+x_{8} g_{-87}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x8+x4x11+x6x136)h8+(x1x8+x2x9+x3x10+x4x11+x5x12+x6x1312)h7+(2x1x8+x2x9+2x3x10+x4x11+x5x12+x6x13+x7x1417)h6+(3x1x8+x2x9+2x3x10+x4x11+2x5x12+x6x13+2x7x1422)h5+(3x1x8+2x2x9+2x3x10+2x4x11+2x5x12+x6x13+3x7x1427)h4+(2x1x8+2x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1418)h3+(x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1414)h2+(x1x8+x2x9+x3x10+x6x13+x7x149)h1[e,f] - h = \left(x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} -6\right)h_{8}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} -12\right)h_{7}+\left(2x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -17\right)h_{6}+\left(3x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -22\right)h_{5}+\left(3x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +3x_{7} x_{14} -27\right)h_{4}+\left(2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -18\right)h_{3}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -14\right)h_{2}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -9\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x2x9+x3x10+x6x13+x7x149=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1414=02x1x8+2x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1418=03x1x8+2x2x9+2x3x10+2x4x11+2x5x12+x6x13+3x7x1427=03x1x8+x2x9+2x3x10+x4x11+2x5x12+x6x13+2x7x1422=02x1x8+x2x9+2x3x10+x4x11+x5x12+x6x13+x7x1417=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x1312=0x1x8+x4x11+x6x136=0\begin{array}{rcl}x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -9&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -14&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -18&=&0\\3x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +3x_{7} x_{14} -27&=&0\\3x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -22&=&0\\2x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -17&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} -12&=&0\\x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} -6&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=6h8+12h7+17h6+22h5+27h4+18h3+14h2+9h1e=x1g87+x7g69+x3g65+x2g58+x4g56+x6g55+x5g54f=g54+g55+g56+g58+g65+g69+g87\begin{array}{rcl}h&=&6h_{8}+12h_{7}+17h_{6}+22h_{5}+27h_{4}+18h_{3}+14h_{2}+9h_{1}\\e&=&x_{1} g_{87}+x_{7} g_{69}+x_{3} g_{65}+x_{2} g_{58}+x_{4} g_{56}+x_{6} g_{55}+x_{5} g_{54}\\f&=&g_{-54}+g_{-55}+g_{-56}+g_{-58}+g_{-65}+g_{-69}+g_{-87}\end{array}
Matrix form of the system we are trying to solve: (11100111111112221111232222133121212212111111111101001010)[col. vect.]=(91418272217126)\begin{pmatrix}1 & 1 & 1 & 0 & 0 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 2\\ 2 & 2 & 1 & 1 & 1 & 1 & 2\\ 3 & 2 & 2 & 2 & 2 & 1 & 3\\ 3 & 1 & 2 & 1 & 2 & 1 & 2\\ 2 & 1 & 2 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 1 & 0 & 0 & 1 & 0 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}9\\ 14\\ 18\\ 27\\ 22\\ 17\\ 12\\ 6\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=6h8+12h7+17h6+22h5+27h4+18h3+14h2+9h1e=x1g87+x7g69+x3g65+x2g58+x4g56+x6g55+x5g54f=x12g54+x13g55+x11g56+x9g58+x10g65+x14g69+x8g87\begin{array}{rcl}h&=&6h_{8}+12h_{7}+17h_{6}+22h_{5}+27h_{4}+18h_{3}+14h_{2}+9h_{1}\\ e&=&x_{1} g_{87}+x_{7} g_{69}+x_{3} g_{65}+x_{2} g_{58}+x_{4} g_{56}+x_{6} g_{55}+x_{5} g_{54}\\ f&=&x_{12} g_{-54}+x_{13} g_{-55}+x_{11} g_{-56}+x_{9} g_{-58}+x_{10} g_{-65}+x_{14} g_{-69}+x_{8} g_{-87}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x2x9+x3x10+x6x13+x7x149=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1414=02x1x8+2x2x9+x3x10+x4x11+x5x12+x6x13+2x7x1418=03x1x8+2x2x9+2x3x10+2x4x11+2x5x12+x6x13+3x7x1427=03x1x8+x2x9+2x3x10+x4x11+2x5x12+x6x13+2x7x1422=02x1x8+x2x9+2x3x10+x4x11+x5x12+x6x13+x7x1417=0x1x8+x2x9+x3x10+x4x11+x5x12+x6x1312=0x1x8+x4x11+x6x136=0\begin{array}{rcl}x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -9&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -14&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -18&=&0\\3x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +3x_{7} x_{14} -27&=&0\\3x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +2x_{7} x_{14} -22&=&0\\2x_{1} x_{8} +x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -17&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} -12&=&0\\x_{1} x_{8} +x_{4} x_{11} +x_{6} x_{13} -6&=&0\\\end{array}

A112A^{12}_1
h-characteristic: (0, 0, 1, 0, 0, 0, 0, 1)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A3+2A1A^{1}_3+2A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V6ψ+6V5ψ+11V4ψ+16V3ψ+15V2ψ+14Vψ+13V0V_{6\psi}+6V_{5\psi}+11V_{4\psi}+16V_{3\psi}+15V_{2\psi}+14V_{\psi}+13V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=6h8+11h7+16h6+21h5+26h4+18h3+13h2+9h1e=g81+3g79+4g69+g58+3g43f=g43+g58+g69+g79+g81\begin{array}{rcl}h&=&6h_{8}+11h_{7}+16h_{6}+21h_{5}+26h_{4}+18h_{3}+13h_{2}+9h_{1}\\ e&=&g_{81}+3g_{79}+4g_{69}+g_{58}+3g_{43}\\ f&=&g_{-43}+g_{-58}+g_{-69}+g_{-79}+g_{-81}\end{array}
Lie brackets of the above elements.
h=6h8+11h7+16h6+21h5+26h4+18h3+13h2+9h1e=g81+3g79+4g69+g58+3g43f=g43+g58+g69+g79+g81\begin{array}{rcl}[e, f]&=&6h_{8}+11h_{7}+16h_{6}+21h_{5}+26h_{4}+18h_{3}+13h_{2}+9h_{1}\\ [h, e]&=&2g_{81}+6g_{79}+8g_{69}+2g_{58}+6g_{43}\\ [h, f]&=&-2g_{-43}-2g_{-58}-2g_{-69}-2g_{-79}-2g_{-81}\end{array}
Centralizer type: B2+A12B_2+A^{2}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 13): g6g_{-6}, g4g_{-4}, h4h_{4}, h6h_{6}, h7+h5+h2h1h_{7}+h_{5}+h_{2}-h_{1}, g1g34g_{1}-g_{-34}, g4g_{4}, g5+g20g_{5}+g_{-20}, g6g_{6}, g12g13g_{12}-g_{-13}, g13g12g_{13}-g_{-12}, g20+g5g_{20}+g_{-5}, g34g1g_{34}-g_{-1}
Basis of centralizer intersected with cartan (dimension: 3): h4-h_{4}, h7+h5+h2h1h_{7}+h_{5}+h_{2}-h_{1}, h6-h_{6}
Cartan of centralizer (dimension: 3): h4-h_{4}, h7+h5+h2h1h_{7}+h_{5}+h_{2}-h_{1}, h6-h_{6}
Cartan-generating semisimple element: 11h7+9h6+11h5h4+11h211h111h_{7}+9h_{6}+11h_{5}-h_{4}+11h_{2}-11h_{1}
adjoint action: (400000000000002400000000000000000000000000000000000000000000000000000002200000000000002400000000000001400000000000004000000000000010000000000000010000000000000014000000000000022)\begin{pmatrix}4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 24 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -22 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -24 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -14 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 22\\ \end{pmatrix}
Characteristic polynomial ad H: x131372x11+633840x9113090368x7+7116903424x587426662400x3x^{13}-1372x^{11}+633840x^9-113090368x^7+7116903424x^5-87426662400x^3
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -24)(x -22)(x -14)(x -10)(x -4)(x +4)(x +10)(x +14)(x +22)(x +24)
Eigenvalues of ad H: 00, 2424, 2222, 1414, 1010, 44, 4-4, 10-10, 14-14, 22-22, 24-24
13 eigenvectors of ad H: 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,0,0,0,1), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,0,0,0,1,0,0), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,1,0,0,0,0,0,0)
Centralizer type: B^{1}_2+A^{2}_1
Reductive components (2 total):
Scalar product computed: (160160160130)\begin{pmatrix}1/60 & -1/60\\ -1/60 & 1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
h6h4h_{6}-h_{4}
matching e: g13g12g_{13}-g_{-12}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2000000000000020000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000200000000000002000000000000020000000000000000000000000000)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h6-h_{6}
matching e: g6g_{-6}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000200000000000001000000000000010000000000000100000000000000)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h6h4h_{6}-h_{4}
matching e: g13g12g_{13}-g_{-12}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2000000000000020000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000200000000000002000000000000020000000000000000000000000000)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h6-h_{6}
matching e: g6g_{-6}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000200000000000001000000000000010000000000000100000000000000)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240120120120)\begin{pmatrix}240 & -120\\ -120 & 120\\ \end{pmatrix}

Scalar product computed: (160)\begin{pmatrix}1/60\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h7+h6+h5+h4+h2h1h_{7}+h_{6}+h_{5}+h_{4}+h_{2}-h_{1}
matching e: g34g1g_{34}-g_{-1}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h7+h6+h5+h4+h2h1h_{7}+h_{6}+h_{5}+h_{4}+h_{2}-h_{1}
matching e: g34g1g_{34}-g_{-1}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240)\begin{pmatrix}240\\ \end{pmatrix}
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Unknown elements.
h=6h8+11h7+16h6+21h5+26h4+18h3+13h2+9h1e=x4g81+x1g79+x2g69+x5g58+x3g43e=x8g43+x10g58+x7g69+x6g79+x9g81\begin{array}{rcl}h&=&6h_{8}+11h_{7}+16h_{6}+21h_{5}+26h_{4}+18h_{3}+13h_{2}+9h_{1}\\ e&=&x_{4} g_{81}+x_{1} g_{79}+x_{2} g_{69}+x_{5} g_{58}+x_{3} g_{43}\\ f&=&x_{8} g_{-43}+x_{10} g_{-58}+x_{7} g_{-69}+x_{6} g_{-79}+x_{9} g_{-81}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x6+x3x86)h8+(2x1x6+x3x8+x4x9+x5x1011)h7+(2x1x6+x2x7+x3x8+2x4x9+x5x1016)h6+(2x1x6+2x2x7+x3x8+3x4x9+x5x1021)h5+(2x1x6+3x2x7+x3x8+3x4x9+2x5x1026)h4+(x1x6+2x2x7+x3x8+2x4x9+2x5x1018)h3+(x1x6+2x2x7+x4x9+x5x1013)h2+(x1x6+x2x7+x4x9+x5x109)h1[e,f] - h = \left(x_{1} x_{6} +x_{3} x_{8} -6\right)h_{8}+\left(2x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -11\right)h_{7}+\left(2x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +2x_{4} x_{9} +x_{5} x_{10} -16\right)h_{6}+\left(2x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +3x_{4} x_{9} +x_{5} x_{10} -21\right)h_{5}+\left(2x_{1} x_{6} +3x_{2} x_{7} +x_{3} x_{8} +3x_{4} x_{9} +2x_{5} x_{10} -26\right)h_{4}+\left(x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -18\right)h_{3}+\left(x_{1} x_{6} +2x_{2} x_{7} +x_{4} x_{9} +x_{5} x_{10} -13\right)h_{2}+\left(x_{1} x_{6} +x_{2} x_{7} +x_{4} x_{9} +x_{5} x_{10} -9\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x6+x2x7+x4x9+x5x109=0x1x6+2x2x7+x4x9+x5x1013=0x1x6+2x2x7+x3x8+2x4x9+2x5x1018=02x1x6+3x2x7+x3x8+3x4x9+2x5x1026=02x1x6+2x2x7+x3x8+3x4x9+x5x1021=02x1x6+x2x7+x3x8+2x4x9+x5x1016=02x1x6+x3x8+x4x9+x5x1011=0x1x6+x3x86=0\begin{array}{rcl}x_{1} x_{6} +x_{2} x_{7} +x_{4} x_{9} +x_{5} x_{10} -9&=&0\\x_{1} x_{6} +2x_{2} x_{7} +x_{4} x_{9} +x_{5} x_{10} -13&=&0\\x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -18&=&0\\2x_{1} x_{6} +3x_{2} x_{7} +x_{3} x_{8} +3x_{4} x_{9} +2x_{5} x_{10} -26&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +3x_{4} x_{9} +x_{5} x_{10} -21&=&0\\2x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +2x_{4} x_{9} +x_{5} x_{10} -16&=&0\\2x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -11&=&0\\x_{1} x_{6} +x_{3} x_{8} -6&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=6h8+11h7+16h6+21h5+26h4+18h3+13h2+9h1e=x4g81+x1g79+x2g69+x5g58+x3g43f=g43+g58+g69+g79+g81\begin{array}{rcl}h&=&6h_{8}+11h_{7}+16h_{6}+21h_{5}+26h_{4}+18h_{3}+13h_{2}+9h_{1}\\e&=&x_{4} g_{81}+x_{1} g_{79}+x_{2} g_{69}+x_{5} g_{58}+x_{3} g_{43}\\f&=&g_{-43}+g_{-58}+g_{-69}+g_{-79}+g_{-81}\end{array}
Matrix form of the system we are trying to solve: (1101112011121222313222131211212011110100)[col. vect.]=(91318262116116)\begin{pmatrix}1 & 1 & 0 & 1 & 1\\ 1 & 2 & 0 & 1 & 1\\ 1 & 2 & 1 & 2 & 2\\ 2 & 3 & 1 & 3 & 2\\ 2 & 2 & 1 & 3 & 1\\ 2 & 1 & 1 & 2 & 1\\ 2 & 0 & 1 & 1 & 1\\ 1 & 0 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}9\\ 13\\ 18\\ 26\\ 21\\ 16\\ 11\\ 6\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=6h8+11h7+16h6+21h5+26h4+18h3+13h2+9h1e=x4g81+x1g79+x2g69+x5g58+x3g43f=x8g43+x10g58+x7g69+x6g79+x9g81\begin{array}{rcl}h&=&6h_{8}+11h_{7}+16h_{6}+21h_{5}+26h_{4}+18h_{3}+13h_{2}+9h_{1}\\ e&=&x_{4} g_{81}+x_{1} g_{79}+x_{2} g_{69}+x_{5} g_{58}+x_{3} g_{43}\\ f&=&x_{8} g_{-43}+x_{10} g_{-58}+x_{7} g_{-69}+x_{6} g_{-79}+x_{9} g_{-81}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x6+x2x7+x4x9+x5x109=0x1x6+2x2x7+x4x9+x5x1013=0x1x6+2x2x7+x3x8+2x4x9+2x5x1018=02x1x6+3x2x7+x3x8+3x4x9+2x5x1026=02x1x6+2x2x7+x3x8+3x4x9+x5x1021=02x1x6+x2x7+x3x8+2x4x9+x5x1016=02x1x6+x3x8+x4x9+x5x1011=0x1x6+x3x86=0\begin{array}{rcl}x_{1} x_{6} +x_{2} x_{7} +x_{4} x_{9} +x_{5} x_{10} -9&=&0\\x_{1} x_{6} +2x_{2} x_{7} +x_{4} x_{9} +x_{5} x_{10} -13&=&0\\x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -18&=&0\\2x_{1} x_{6} +3x_{2} x_{7} +x_{3} x_{8} +3x_{4} x_{9} +2x_{5} x_{10} -26&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +3x_{4} x_{9} +x_{5} x_{10} -21&=&0\\2x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +2x_{4} x_{9} +x_{5} x_{10} -16&=&0\\2x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -11&=&0\\x_{1} x_{6} +x_{3} x_{8} -6&=&0\\\end{array}

A112A^{12}_1
h-characteristic: (0, 0, 0, 0, 0, 0, 2, 0)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: 3A23A^{1}_2 Containing regular semisimple subalgebra number 2: A3+2A1A^{1}_3+2A^{1}_1 Containing regular semisimple subalgebra number 3: D4D^{1}_4
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 2V6ψ+25V4ψ+27V2ψ+28V02V_{6\psi}+25V_{4\psi}+27V_{2\psi}+28V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=6h8+12h7+16h6+20h5+24h4+16h3+12h2+8h1e=2g101+2g75+2g60+2g43+2g42+2g40f=g40+g42+g43+g60+g75+g101\begin{array}{rcl}h&=&6h_{8}+12h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ e&=&2g_{101}+2g_{75}+2g_{60}+2g_{43}+2g_{42}+2g_{40}\\ f&=&g_{-40}+g_{-42}+g_{-43}+g_{-60}+g_{-75}+g_{-101}\end{array}
Lie brackets of the above elements.
h=6h8+12h7+16h6+20h5+24h4+16h3+12h2+8h1e=2g101+2g75+2g60+2g43+2g42+2g40f=g40+g42+g43+g60+g75+g101\begin{array}{rcl}[e, f]&=&6h_{8}+12h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ [h, e]&=&4g_{101}+4g_{75}+4g_{60}+4g_{43}+4g_{42}+4g_{40}\\ [h, f]&=&-2g_{-40}-2g_{-42}-2g_{-43}-2g_{-60}-2g_{-75}-2g_{-101}\end{array}
Centralizer type: D4D_4
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Unknown elements.
h=6h8+12h7+16h6+20h5+24h4+16h3+12h2+8h1e=x1g101+x3g75+x2g60+x4g43+x6g42+x5g40e=x11g40+x12g42+x10g43+x8g60+x9g75+x7g101\begin{array}{rcl}h&=&6h_{8}+12h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ e&=&x_{1} g_{101}+x_{3} g_{75}+x_{2} g_{60}+x_{4} g_{43}+x_{6} g_{42}+x_{5} g_{40}\\ f&=&x_{11} g_{-40}+x_{12} g_{-42}+x_{10} g_{-43}+x_{8} g_{-60}+x_{9} g_{-75}+x_{7} g_{-101}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x7+x4x10+x6x126)h8+(x1x7+x2x8+x3x9+x4x10+x5x11+x6x1212)h7+(2x1x7+2x2x8+x3x9+x4x10+x5x11+x6x1216)h6+(3x1x7+2x2x8+2x3x9+x4x10+x5x11+x6x1220)h5+(4x1x7+2x2x8+3x3x9+x4x10+x5x11+x6x1224)h4+(3x1x7+x2x8+2x3x9+x4x10+x5x1116)h3+(2x1x7+x2x8+2x3x9+x6x1212)h2+(2x1x7+x3x9+x5x118)h1[e,f] - h = \left(x_{1} x_{7} +x_{4} x_{10} +x_{6} x_{12} -6\right)h_{8}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -12\right)h_{7}+\left(2x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -16\right)h_{6}+\left(3x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -20\right)h_{5}+\left(4x_{1} x_{7} +2x_{2} x_{8} +3x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -24\right)h_{4}+\left(3x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} -16\right)h_{3}+\left(2x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{6} x_{12} -12\right)h_{2}+\left(2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -8\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
2x1x7+x3x9+x5x118=02x1x7+x2x8+2x3x9+x6x1212=03x1x7+x2x8+2x3x9+x4x10+x5x1116=04x1x7+2x2x8+3x3x9+x4x10+x5x11+x6x1224=03x1x7+2x2x8+2x3x9+x4x10+x5x11+x6x1220=02x1x7+2x2x8+x3x9+x4x10+x5x11+x6x1216=0x1x7+x2x8+x3x9+x4x10+x5x11+x6x1212=0x1x7+x4x10+x6x126=0\begin{array}{rcl}2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -8&=&0\\2x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{6} x_{12} -12&=&0\\3x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} -16&=&0\\4x_{1} x_{7} +2x_{2} x_{8} +3x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -24&=&0\\3x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -16&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -12&=&0\\x_{1} x_{7} +x_{4} x_{10} +x_{6} x_{12} -6&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=6h8+12h7+16h6+20h5+24h4+16h3+12h2+8h1e=x1g101+x3g75+x2g60+x4g43+x6g42+x5g40f=g40+g42+g43+g60+g75+g101\begin{array}{rcl}h&=&6h_{8}+12h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\e&=&x_{1} g_{101}+x_{3} g_{75}+x_{2} g_{60}+x_{4} g_{43}+x_{6} g_{42}+x_{5} g_{40}\\f&=&g_{-40}+g_{-42}+g_{-43}+g_{-60}+g_{-75}+g_{-101}\end{array}
Matrix form of the system we are trying to solve: (201010212001312110423111322111221111111111100101)[col. vect.]=(81216242016126)\begin{pmatrix}2 & 0 & 1 & 0 & 1 & 0\\ 2 & 1 & 2 & 0 & 0 & 1\\ 3 & 1 & 2 & 1 & 1 & 0\\ 4 & 2 & 3 & 1 & 1 & 1\\ 3 & 2 & 2 & 1 & 1 & 1\\ 2 & 2 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 0 & 0 & 1 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 12\\ 16\\ 24\\ 20\\ 16\\ 12\\ 6\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=6h8+12h7+16h6+20h5+24h4+16h3+12h2+8h1e=x1g101+x3g75+x2g60+x4g43+x6g42+x5g40f=x11g40+x12g42+x10g43+x8g60+x9g75+x7g101\begin{array}{rcl}h&=&6h_{8}+12h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ e&=&x_{1} g_{101}+x_{3} g_{75}+x_{2} g_{60}+x_{4} g_{43}+x_{6} g_{42}+x_{5} g_{40}\\ f&=&x_{11} g_{-40}+x_{12} g_{-42}+x_{10} g_{-43}+x_{8} g_{-60}+x_{9} g_{-75}+x_{7} g_{-101}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
2x1x7+x3x9+x5x118=02x1x7+x2x8+2x3x9+x6x1212=03x1x7+x2x8+2x3x9+x4x10+x5x1116=04x1x7+2x2x8+3x3x9+x4x10+x5x11+x6x1224=03x1x7+2x2x8+2x3x9+x4x10+x5x11+x6x1220=02x1x7+2x2x8+x3x9+x4x10+x5x11+x6x1216=0x1x7+x2x8+x3x9+x4x10+x5x11+x6x1212=0x1x7+x4x10+x6x126=0\begin{array}{rcl}2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -8&=&0\\2x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{6} x_{12} -12&=&0\\3x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} -16&=&0\\4x_{1} x_{7} +2x_{2} x_{8} +3x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -24&=&0\\3x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -16&=&0\\x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -12&=&0\\x_{1} x_{7} +x_{4} x_{10} +x_{6} x_{12} -6&=&0\\\end{array}

A111A^{11}_1
h-characteristic: (0, 0, 0, 0, 0, 1, 0, 1)
Length of the weight dual to h: 22
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A3+A1A^{1}_3+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V6ψ+2V5ψ+15V4ψ+18V3ψ+10V2ψ+14Vψ+24V0V_{6\psi}+2V_{5\psi}+15V_{4\psi}+18V_{3\psi}+10V_{2\psi}+14V_{\psi}+24V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=6h8+11h7+16h6+20h5+24h4+16h3+12h2+8h1e=3g82+4g81+g80+3g22f=g22+g80+g81+g82\begin{array}{rcl}h&=&6h_{8}+11h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ e&=&3g_{82}+4g_{81}+g_{80}+3g_{22}\\ f&=&g_{-22}+g_{-80}+g_{-81}+g_{-82}\end{array}
Lie brackets of the above elements.
h=6h8+11h7+16h6+20h5+24h4+16h3+12h2+8h1e=3g82+4g81+g80+3g22f=g22+g80+g81+g82\begin{array}{rcl}[e, f]&=&6h_{8}+11h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ [h, e]&=&6g_{82}+8g_{81}+2g_{80}+6g_{22}\\ [h, f]&=&-2g_{-22}-2g_{-80}-2g_{-81}-2g_{-82}\end{array}
Centralizer type: B3+A1B_3+A_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 24): g16g_{-16}, g11g_{-11}, g9g_{-9}, g7g_{-7}, g4g_{-4}, g3g_{-3}, g1g_{-1}, h1h_{1}, h3h_{3}, h4h_{4}, h7h_{7}, g1g_{1}, g3g_{3}, g4g_{4}, g7g_{7}, g9g_{9}, g11g_{11}, g16g_{16}, g18g44g_{18}-g_{-44}, g25+g37g_{25}+g_{-37}, g30g31g_{30}-g_{-31}, g31g30g_{31}-g_{-30}, g37+g25g_{37}+g_{-25}, g44g18g_{44}-g_{-18}
Basis of centralizer intersected with cartan (dimension: 4): h4-h_{4}, h3-h_{3}, h1-h_{1}, h7-h_{7}
Cartan of centralizer (dimension: 4): h1-h_{1}, h7-h_{7}, h4-h_{4}, h3-h_{3}
Cartan-generating semisimple element: 7h7h411h3+9h17h_{7}-h_{4}-11h_{3}+9h_{1}
adjoint action: (80000000000000000000000002100000000000000000000000010000000000000000000000001400000000000000000000000090000000000000000000000003000000000000000000000000029000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000029000000000000000000000000300000000000000000000000009000000000000000000000000140000000000000000000000001000000000000000000000000210000000000000000000000008000000000000000000000000110000000000000000000000001900000000000000000000000010000000000000000000000000100000000000000000000000001900000000000000000000000011)\begin{pmatrix}-8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 21 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 30 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -29 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 29 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -30 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -21 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -19 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 19 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -11\\ \end{pmatrix}
Characteristic polynomial ad H: x243106x22+3865719x202508061544x18+932467254575x16206377397552586x14+27397665443609161x122128206127842387724x10+89025856368108579504x81568387575821170664000x6+1481462733362895360000x4x^{24}-3106x^{22}+3865719x^{20}-2508061544x^{18}+932467254575x^{16}-206377397552586x^{14}+27397665443609161x^{12}-2128206127842387724x^{10}+89025856368108579504x^8-1568387575821170664000x^6+1481462733362895360000x^4
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -30)(x -29)(x -21)(x -19)(x -14)(x -11)(x -10)(x -9)(x -8)(x -1)(x +1)(x +8)(x +9)(x +10)(x +11)(x +14)(x +19)(x +21)(x +29)(x +30)
Eigenvalues of ad H: 00, 3030, 2929, 2121, 1919, 1414, 1111, 1010, 99, 88, 11, 1-1, 8-8, 9-9, 10-10, 11-11, 14-14, 19-19, 21-21, 29-29, 30-30
24 eigenvectors of ad H: 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1), 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)
Centralizer type: B^{1}_3+A^{1}_1
Reductive components (2 total):
Scalar product computed: (16016001601301600160130)\begin{pmatrix}1/60 & -1/60 & 0\\ -1/60 & 1/30 & -1/60\\ 0 & -1/60 & 1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (3 total):
h4+h1-h_{4}+h_{1}
matching e: g30g31g_{30}-g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000020000000000000000000000002000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000200000000000000000000000020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000200000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h3h1-h_{3}-h_{1}
matching e: g9g_{-9}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (100000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000010000000000000000000000001000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000100000000000000000000000010000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000001000000000000000000000000100000000000000000000000000000000000000000000000001000000000000000000000000100000000000000000000000000000000000000000000000001)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
h4+h3+h1h_{4}+h_{3}+h_{1}
matching e: g16g_{16}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000010000000000000000000000001000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000100000000000000000000000010000000000000000000000002000000000000000000000000100000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000001)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h4+h1-h_{4}+h_{1}
matching e: g30g31g_{30}-g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000020000000000000000000000002000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000200000000000000000000000020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000200000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h3h1-h_{3}-h_{1}
matching e: g9g_{-9}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (100000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000010000000000000000000000001000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000100000000000000000000000010000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000001000000000000000000000000100000000000000000000000000000000000000000000000001000000000000000000000000100000000000000000000000000000000000000000000000001)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
h4+h3+h1h_{4}+h_{3}+h_{1}
matching e: g16g_{16}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000010000000000000000000000001000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000100000000000000000000000010000000000000000000000002000000000000000000000000100000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000001)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
Elements in Cartan dual to root system: (1, 2, 1), (-1, -2, -1), (1, 1, 1), (-1, -1, -1), (1, 1, 0), (-1, -1, 0), (1, 2, 2), (-1, -2, -2), (1, 2, 0), (-1, -2, 0), (1, 0, 0), (-1, 0, 0), (0, 1, 1), (0, -1, -1), (0, 0, 1), (0, 0, -1), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240120012012060060120)\begin{pmatrix}240 & -120 & 0\\ -120 & 120 & -60\\ 0 & -60 & 120\\ \end{pmatrix}

Scalar product computed: (130)\begin{pmatrix}1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h7h_{7}
matching e: g7g_{7}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h7h_{7}
matching e: g7g_{7}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120)\begin{pmatrix}120\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=6h8+11h7+16h6+20h5+24h4+16h3+12h2+8h1e=x1g82+x2g81+x4g80+x3g22e=x7g22+x8g80+x6g81+x5g82\begin{array}{rcl}h&=&6h_{8}+11h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ e&=&x_{1} g_{82}+x_{2} g_{81}+x_{4} g_{80}+x_{3} g_{22}\\ f&=&x_{7} g_{-22}+x_{8} g_{-80}+x_{6} g_{-81}+x_{5} g_{-82}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x5+x3x76)h8+(x1x5+x2x6+x3x7+x4x811)h7+(x1x5+2x2x6+x3x7+2x4x816)h6+(2x1x5+3x2x6+2x4x820)h5+(3x1x5+3x2x6+3x4x824)h4+(2x1x5+2x2x6+2x4x816)h3+(2x1x5+x2x6+2x4x812)h2+(x1x5+x2x6+x4x88)h1[e,f] - h = \left(x_{1} x_{5} +x_{3} x_{7} -6\right)h_{8}+\left(x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -11\right)h_{7}+\left(x_{1} x_{5} +2x_{2} x_{6} +x_{3} x_{7} +2x_{4} x_{8} -16\right)h_{6}+\left(2x_{1} x_{5} +3x_{2} x_{6} +2x_{4} x_{8} -20\right)h_{5}+\left(3x_{1} x_{5} +3x_{2} x_{6} +3x_{4} x_{8} -24\right)h_{4}+\left(2x_{1} x_{5} +2x_{2} x_{6} +2x_{4} x_{8} -16\right)h_{3}+\left(2x_{1} x_{5} +x_{2} x_{6} +2x_{4} x_{8} -12\right)h_{2}+\left(x_{1} x_{5} +x_{2} x_{6} +x_{4} x_{8} -8\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x5+x2x6+x4x88=02x1x5+x2x6+2x4x812=02x1x5+2x2x6+2x4x816=03x1x5+3x2x6+3x4x824=02x1x5+3x2x6+2x4x820=0x1x5+2x2x6+x3x7+2x4x816=0x1x5+x2x6+x3x7+x4x811=0x1x5+x3x76=0\begin{array}{rcl}x_{1} x_{5} +x_{2} x_{6} +x_{4} x_{8} -8&=&0\\2x_{1} x_{5} +x_{2} x_{6} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{4} x_{8} -16&=&0\\3x_{1} x_{5} +3x_{2} x_{6} +3x_{4} x_{8} -24&=&0\\2x_{1} x_{5} +3x_{2} x_{6} +2x_{4} x_{8} -20&=&0\\x_{1} x_{5} +2x_{2} x_{6} +x_{3} x_{7} +2x_{4} x_{8} -16&=&0\\x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -11&=&0\\x_{1} x_{5} +x_{3} x_{7} -6&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=6h8+11h7+16h6+20h5+24h4+16h3+12h2+8h1e=x1g82+x2g81+x4g80+x3g22f=g22+g80+g81+g82\begin{array}{rcl}h&=&6h_{8}+11h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\e&=&x_{1} g_{82}+x_{2} g_{81}+x_{4} g_{80}+x_{3} g_{22}\\f&=&g_{-22}+g_{-80}+g_{-81}+g_{-82}\end{array}
Matrix form of the system we are trying to solve: (11012102220233032302121211111010)[col. vect.]=(81216242016116)\begin{pmatrix}1 & 1 & 0 & 1\\ 2 & 1 & 0 & 2\\ 2 & 2 & 0 & 2\\ 3 & 3 & 0 & 3\\ 2 & 3 & 0 & 2\\ 1 & 2 & 1 & 2\\ 1 & 1 & 1 & 1\\ 1 & 0 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 12\\ 16\\ 24\\ 20\\ 16\\ 11\\ 6\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=6h8+11h7+16h6+20h5+24h4+16h3+12h2+8h1e=x1g82+x2g81+x4g80+x3g22f=x7g22+x8g80+x6g81+x5g82\begin{array}{rcl}h&=&6h_{8}+11h_{7}+16h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ e&=&x_{1} g_{82}+x_{2} g_{81}+x_{4} g_{80}+x_{3} g_{22}\\ f&=&x_{7} g_{-22}+x_{8} g_{-80}+x_{6} g_{-81}+x_{5} g_{-82}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x5+x2x6+x4x88=02x1x5+x2x6+2x4x812=02x1x5+2x2x6+2x4x816=03x1x5+3x2x6+3x4x824=02x1x5+3x2x6+2x4x820=0x1x5+2x2x6+x3x7+2x4x816=0x1x5+x2x6+x3x7+x4x811=0x1x5+x3x76=0\begin{array}{rcl}x_{1} x_{5} +x_{2} x_{6} +x_{4} x_{8} -8&=&0\\2x_{1} x_{5} +x_{2} x_{6} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{4} x_{8} -16&=&0\\3x_{1} x_{5} +3x_{2} x_{6} +3x_{4} x_{8} -24&=&0\\2x_{1} x_{5} +3x_{2} x_{6} +2x_{4} x_{8} -20&=&0\\x_{1} x_{5} +2x_{2} x_{6} +x_{3} x_{7} +2x_{4} x_{8} -16&=&0\\x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -11&=&0\\x_{1} x_{5} +x_{3} x_{7} -6&=&0\\\end{array}

A110A^{10}_1
h-characteristic: (1, 0, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A3A^{1}_3
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V6ψ+11V4ψ+32V3ψ+V2ψ+55V0V_{6\psi}+11V_{4\psi}+32V_{3\psi}+V_{2\psi}+55V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=6h8+10h7+14h6+18h5+22h4+15h3+11h2+8h1e=4g97+3g74+3g8f=g8+g74+g97\begin{array}{rcl}h&=&6h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\\ e&=&4g_{97}+3g_{74}+3g_{8}\\ f&=&g_{-8}+g_{-74}+g_{-97}\end{array}
Lie brackets of the above elements.
h=6h8+10h7+14h6+18h5+22h4+15h3+11h2+8h1e=4g97+3g74+3g8f=g8+g74+g97\begin{array}{rcl}[e, f]&=&6h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\\ [h, e]&=&8g_{97}+6g_{74}+6g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-74}-2g_{-97}\end{array}
Centralizer type: B5B_5
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 55): g46g_{-46}, g39g_{-39}, g33g_{-33}, g31g_{-31}, g27g_{-27}, g26g_{-26}, g25g_{-25}, g20g_{-20}, g19g_{-19}, g18g_{-18}, g17g_{-17}, g13g_{-13}, g12g_{-12}, g11g_{-11}, g10g_{-10}, g6g_{-6}, g5g_{-5}, g4g_{-4}, g3g_{-3}, g2g_{-2}, h2h_{2}, h3h_{3}, h4h_{4}, h5h_{5}, h6h_{6}, g2g_{2}, g3g_{3}, g4g_{4}, g5g_{5}, g6g_{6}, g7g60g_{7}-g_{-60}, g10g_{10}, g11g_{11}, g12g_{12}, g13g_{13}, g14+g54g_{14}+g_{-54}, g17g_{17}, g18g_{18}, g19g_{19}, g20g_{20}, g21g48g_{21}-g_{-48}, g25g_{25}, g26g_{26}, g27g_{27}, g28+g41g_{28}+g_{-41}, g31g_{31}, g33g_{33}, g34g35g_{34}-g_{-35}, g35g34g_{35}-g_{-34}, g39g_{39}, g41+g28g_{41}+g_{-28}, g46g_{46}, g48g21g_{48}-g_{-21}, g54+g14g_{54}+g_{-14}, g60g7g_{60}-g_{-7}
Basis of centralizer intersected with cartan (dimension: 5): h6-h_{6}, h5-h_{5}, h4-h_{4}, h3-h_{3}, h2-h_{2}
Cartan of centralizer (dimension: 5): h6-h_{6}, h5-h_{5}, h4-h_{4}, h3-h_{3}, h2-h_{2}
Cartan-generating semisimple element: h611h5+9h4+7h34h2-h_{6}-11h_{5}+9h_{4}+7h_{3}-4h_{2}
adjoint action: 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0 & 0 & 0 & 0 & 19 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -11 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 20 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Characteristic polynomial ad H: x555922x53+15957243x5126026191744x49+28837243366425x4723094669408653718x45+13886578848706275291x436428068430617948173804x41+2330231696547426994320747x39669378358899689968426206350x37+153580383979319146083834901833x3528281338559820339960572682103016x33+4189204541649834789657311323123179x31499034592079519205102017649307154362x29+47677711396016753922879207445251341025x273633491217486234462501389076305868859116x25+218988792774023264383410327808919951152272x2310308035683734281364098265589341113256658368x21+372323991350451756162310396967391191890441984x1910065796883243580151212541361434834467427353600x17+196488391514817746330752644785450014059732480000x152621710155245982777380991811370260303809792000000x13+21816780600112751989826695105773504719155200000000x1194603083573219906571511352168704181166080000000000x9+133318093052074134476666273199974793216000000000000x758096864749710722395672046216308326400000000000000x5x^{55}-5922x^{53}+15957243x^{51}-26026191744x^{49}+28837243366425x^{47}-23094669408653718x^{45}+13886578848706275291x^{43}-6428068430617948173804x^{41}+2330231696547426994320747x^{39}-669378358899689968426206350x^{37}+153580383979319146083834901833x^{35}-28281338559820339960572682103016x^{33}+4189204541649834789657311323123179x^{31}-499034592079519205102017649307154362x^{29}+47677711396016753922879207445251341025x^{27}-3633491217486234462501389076305868859116x^{25}+218988792774023264383410327808919951152272x^{23}-10308035683734281364098265589341113256658368x^{21}+372323991350451756162310396967391191890441984x^{19}-10065796883243580151212541361434834467427353600x^{17}+196488391514817746330752644785450014059732480000x^{15}-2621710155245982777380991811370260303809792000000x^{13}+21816780600112751989826695105773504719155200000000x^{11}-94603083573219906571511352168704181166080000000000x^9+133318093052074134476666273199974793216000000000000x^7-58096864749710722395672046216308326400000000000000x^5
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x )(x -31)(x -30)(x -26)(x -21)(x -21)(x -20)(x -19)(x -17)(x -16)(x -14)(x -12)(x -11)(x -11)(x -10)(x -10)(x -10)(x -9)(x -9)(x -7)(x -6)(x -5)(x -5)(x -4)(x -1)(x -1)(x +1)(x +1)(x +4)(x +5)(x +5)(x +6)(x +7)(x +9)(x +9)(x +10)(x +10)(x +10)(x +11)(x +11)(x +12)(x +14)(x +16)(x +17)(x +19)(x +20)(x +21)(x +21)(x +26)(x +30)(x +31)
Eigenvalues of ad H: 00, 3131, 3030, 2626, 2121, 2020, 1919, 1717, 1616, 1414, 1212, 1111, 1010, 99, 77, 66, 55, 44, 11, 1-1, 4-4, 5-5, 6-6, 7-7, 9-9, 10-10, 11-11, 12-12, 14-14, 16-16, 17-17, 19-19, 20-20, 21-21, 26-26, 30-30, 31-31
55 eigenvectors of ad H: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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Centralizer type: B^{1}_5
Reductive components (1 total):
Scalar product computed: (130001600016016000016013001601600013016000160160130)\begin{pmatrix}1/30 & 0 & 0 & -1/60 & 0\\ 0 & 1/60 & -1/60 & 0 & 0\\ 0 & -1/60 & 1/30 & 0 & -1/60\\ -1/60 & 0 & 0 & 1/30 & -1/60\\ 0 & 0 & -1/60 & -1/60 & 1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (5 total):
h4+h2h_{4}+h_{2}
matching e: g10g_{10}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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2h62h52h4h3h2-2h_{6}-2h_{5}-2h_{4}-h_{3}-h_{2}
matching e: g7g60g_{7}-g_{-60}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h6+h5+h4h_{6}+h_{5}+h_{4}
matching e: g20g_{20}
verification: [h,e]2e=0[h,e]-2e=0
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h5+h4+h3h_{5}+h_{4}+h_{3}
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verification: [h,e]2e=0[h,e]-2e=0
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h5h4-h_{5}-h_{4}
matching e: g12g_{-12}
verification: [h,e]2e=0[h,e]-2e=0
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Linear space basis of intersection of centralizer and ambient Cartan:
h4+h2h_{4}+h_{2}
matching e: g10g_{10}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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2h62h52h4h3h2-2h_{6}-2h_{5}-2h_{4}-h_{3}-h_{2}
matching e: g7g60g_{7}-g_{-60}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h6+h5+h4h_{6}+h_{5}+h_{4}
matching e: g20g_{20}
verification: [h,e]2e=0[h,e]-2e=0
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h5+h4+h3h_{5}+h_{4}+h_{3}
matching e: g19g_{19}
verification: [h,e]2e=0[h,e]-2e=0
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h5h4-h_{5}-h_{4}
matching e: g12g_{-12}
verification: [h,e]2e=0[h,e]-2e=0
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& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1, 1, 2, 2, 2), (-1, -1, -2, -2, -2), (1, 1, 2, 1, 2), (-1, -1, -2, -1, -2), (1, 1, 2, 1, 1), (-1, -1, -2, -1, -1), (0, 1, 2, 1, 2), (0, -1, -2, -1, -2), (1, 1, 1, 1, 1), (-1, -1, -1, -1, -1), (2, 1, 2, 2, 2), (-2, -1, -2, -2, -2), (1, 0, 1, 1, 1), (-1, 0, -1, -1, -1), (0, 1, 2, 1, 1), (0, -1, -2, -1, -1), (0, 1, 2, 0, 1), (0, -1, -2, 0, -1), (1, 0, 0, 1, 1), (-1, 0, 0, -1, -1), (0, 1, 1, 1, 1), (0, -1, -1, -1, -1), (0, 1, 1, 0, 1), (0, -1, -1, 0, -1), (0, 1, 2, 2, 2), (0, -1, -2, -2, -2), (0, 1, 2, 0, 2), (0, -1, -2, 0, -2), (0, 0, 1, 1, 1), (0, 0, -1, -1, -1), (1, 0, 0, 1, 0), (-1, 0, 0, -1, 0), (0, 0, 1, 0, 1), (0, 0, -1, 0, -1), (1, 0, 0, 0, 0), (-1, 0, 0, 0, 0), (0, 1, 1, 0, 0), (0, -1, -1, 0, 0), (0, 1, 2, 0, 0), (0, -1, -2, 0, 0), (0, 0, 1, 0, 0), (0, 0, -1, 0, 0), (0, 0, 0, 1, 1), (0, 0, 0, -1, -1), (0, 0, 0, 0, 1), (0, 0, 0, 0, -1), (0, 0, 0, 1, 0), (0, 0, 0, -1, 0), (0, 1, 0, 0, 0), (0, -1, 0, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120006000240120000120120060600012060006060120)\begin{pmatrix}120 & 0 & 0 & -60 & 0\\ 0 & 240 & -120 & 0 & 0\\ 0 & -120 & 120 & 0 & -60\\ -60 & 0 & 0 & 120 & -60\\ 0 & 0 & -60 & -60 & 120\\ \end{pmatrix}
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Unknown elements.
h=6h8+10h7+14h6+18h5+22h4+15h3+11h2+8h1e=x2g97+x1g74+x3g8e=x6g8+x4g74+x5g97\begin{array}{rcl}h&=&6h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\\ e&=&x_{2} g_{97}+x_{1} g_{74}+x_{3} g_{8}\\ f&=&x_{6} g_{-8}+x_{4} g_{-74}+x_{5} g_{-97}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x4+x3x66)h8+(2x1x4+x2x510)h7+(2x1x4+2x2x514)h6+(2x1x4+3x2x518)h5+(2x1x4+4x2x522)h4+(x1x4+3x2x515)h3+(x1x4+2x2x511)h2+(2x2x58)h1[e,f] - h = \left(x_{1} x_{4} +x_{3} x_{6} -6\right)h_{8}+\left(2x_{1} x_{4} +x_{2} x_{5} -10\right)h_{7}+\left(2x_{1} x_{4} +2x_{2} x_{5} -14\right)h_{6}+\left(2x_{1} x_{4} +3x_{2} x_{5} -18\right)h_{5}+\left(2x_{1} x_{4} +4x_{2} x_{5} -22\right)h_{4}+\left(x_{1} x_{4} +3x_{2} x_{5} -15\right)h_{3}+\left(x_{1} x_{4} +2x_{2} x_{5} -11\right)h_{2}+\left(2x_{2} x_{5} -8\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x4+2x2x511=0x1x4+3x2x515=02x1x4+4x2x522=02x1x4+3x2x518=02x1x4+2x2x514=02x1x4+x2x510=0x1x4+x3x66=02x2x58=0\begin{array}{rcl}x_{1} x_{4} +2x_{2} x_{5} -11&=&0\\x_{1} x_{4} +3x_{2} x_{5} -15&=&0\\2x_{1} x_{4} +4x_{2} x_{5} -22&=&0\\2x_{1} x_{4} +3x_{2} x_{5} -18&=&0\\2x_{1} x_{4} +2x_{2} x_{5} -14&=&0\\2x_{1} x_{4} +x_{2} x_{5} -10&=&0\\x_{1} x_{4} +x_{3} x_{6} -6&=&0\\2x_{2} x_{5} -8&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=6h8+10h7+14h6+18h5+22h4+15h3+11h2+8h1e=x2g97+x1g74+x3g8f=g8+g74+g97\begin{array}{rcl}h&=&6h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\\e&=&x_{2} g_{97}+x_{1} g_{74}+x_{3} g_{8}\\f&=&g_{-8}+g_{-74}+g_{-97}\end{array}
Matrix form of the system we are trying to solve: (120130240230220210101020)[col. vect.]=(11152218141068)\begin{pmatrix}1 & 2 & 0\\ 1 & 3 & 0\\ 2 & 4 & 0\\ 2 & 3 & 0\\ 2 & 2 & 0\\ 2 & 1 & 0\\ 1 & 0 & 1\\ 0 & 2 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}11\\ 15\\ 22\\ 18\\ 14\\ 10\\ 6\\ 8\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=6h8+10h7+14h6+18h5+22h4+15h3+11h2+8h1e=x2g97+x1g74+x3g8f=x6g8+x4g74+x5g97\begin{array}{rcl}h&=&6h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\\ e&=&x_{2} g_{97}+x_{1} g_{74}+x_{3} g_{8}\\ f&=&x_{6} g_{-8}+x_{4} g_{-74}+x_{5} g_{-97}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x4+2x2x511=0x1x4+3x2x515=02x1x4+4x2x522=02x1x4+3x2x518=02x1x4+2x2x514=02x1x4+x2x510=0x1x4+x3x66=02x2x58=0\begin{array}{rcl}x_{1} x_{4} +2x_{2} x_{5} -11&=&0\\x_{1} x_{4} +3x_{2} x_{5} -15&=&0\\2x_{1} x_{4} +4x_{2} x_{5} -22&=&0\\2x_{1} x_{4} +3x_{2} x_{5} -18&=&0\\2x_{1} x_{4} +2x_{2} x_{5} -14&=&0\\2x_{1} x_{4} +x_{2} x_{5} -10&=&0\\x_{1} x_{4} +x_{3} x_{6} -6&=&0\\2x_{2} x_{5} -8&=&0\\\end{array}

A110A^{10}_1
h-characteristic: (0, 0, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: 2A2+2A12A^{1}_2+2A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 4V5ψ+10V4ψ+16V3ψ+20V2ψ+20Vψ+10V04V_{5\psi}+10V_{4\psi}+16V_{3\psi}+20V_{2\psi}+20V_{\psi}+10V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=5h8+10h7+15h6+20h5+24h4+16h3+12h2+8h1e=2g91+2g78+2g63+g62+g60+2g59f=g59+g60+g62+g63+g78+g91\begin{array}{rcl}h&=&5h_{8}+10h_{7}+15h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ e&=&2g_{91}+2g_{78}+2g_{63}+g_{62}+g_{60}+2g_{59}\\ f&=&g_{-59}+g_{-60}+g_{-62}+g_{-63}+g_{-78}+g_{-91}\end{array}
Lie brackets of the above elements.
h=5h8+10h7+15h6+20h5+24h4+16h3+12h2+8h1e=2g91+2g78+2g63+g62+g60+2g59f=g59+g60+g62+g63+g78+g91\begin{array}{rcl}[e, f]&=&5h_{8}+10h_{7}+15h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ [h, e]&=&4g_{91}+4g_{78}+4g_{63}+2g_{62}+2g_{60}+4g_{59}\\ [h, f]&=&-2g_{-59}-2g_{-60}-2g_{-62}-2g_{-63}-2g_{-78}-2g_{-91}\end{array}
Centralizer type: B23B^{3}_2
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 10): h7+h3h2h_{7}+h_{3}-h_{2}, h8+h62h3h_{8}+h_{6}-2h_{3}, g8g6+2g1g9g_{8}-g_{6}+2g_{1}-g_{-9}, g912g1+12g612g8g_{9}-1/2g_{-1}+1/2g_{-6}-1/2g_{-8}, g10g7+g11g_{10}-g_{-7}+g_{-11}, g11g7+g10g_{11}-g_{7}+g_{-10}, g16+12g15+12g14+12g23g_{16}+1/2g_{15}+1/2g_{14}+1/2g_{-23}, g17+g4+g22g_{17}+g_{-4}+g_{-22}, g22+g4+g17g_{22}+g_{4}+g_{-17}, g23+12g14+12g15+12g16g_{23}+1/2g_{-14}+1/2g_{-15}+1/2g_{-16}
Basis of centralizer intersected with cartan (dimension: 2): 12h8+12h6h31/2h_{8}+1/2h_{6}-h_{3}, 12h8+h7+12h6h21/2h_{8}+h_{7}+1/2h_{6}-h_{2}
Cartan of centralizer (dimension: 2): 12h8+12h6h31/2h_{8}+1/2h_{6}-h_{3}, 12h8+h7+12h6h21/2h_{8}+h_{7}+1/2h_{6}-h_{2}
Cartan-generating semisimple element: 6h8+11h7+6h6h311h26h_{8}+11h_{7}+6h_{6}-h_{3}-11h_{2}
adjoint action: (0000000000000000000000100000000001000000000010000000000010000000000011000000000012000000000012000000000011)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 11 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -12 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 12 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -11\\ \end{pmatrix}
Characteristic polynomial ad H: x10366x8+44289x61786324x4+1742400x2x^{10}-366x^8+44289x^6-1786324x^4+1742400x^2
Factorization of characteristic polynomial of ad H: (x )(x )(x -12)(x -11)(x -10)(x -1)(x +1)(x +10)(x +11)(x +12)
Eigenvalues of ad H: 00, 1212, 1111, 1010, 11, 1-1, 10-10, 11-11, 12-12
10 eigenvectors of ad H: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,1), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,1,0,0)
Centralizer type: B^{3}_2
Reductive components (1 total):
Scalar product computed: (190118011801180)\begin{pmatrix}1/90 & -1/180\\ -1/180 & 1/180\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
h7+h3h2h_{7}+h_{3}-h_{2}
matching e: g11g7+g10g_{11}-g_{7}+g_{-10}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000100000000001000000000020000000000200000000001000000000000000000000000000000001)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
h8+h62h3h_{8}+h_{6}-2h_{3}
matching e: g8g6+2g1g9g_{8}-g_{6}+2g_{1}-g_{-9}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000200000000002000000000020000000000200000000000000000000020000000000200000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h7+h3h2h_{7}+h_{3}-h_{2}
matching e: g11g7+g10g_{11}-g_{7}+g_{-10}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000100000000001000000000020000000000200000000001000000000000000000000000000000001)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
h8+h62h3h_{8}+h_{6}-2h_{3}
matching e: g8g6+2g1g9g_{8}-g_{6}+2g_{1}-g_{-9}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000200000000002000000000020000000000200000000000000000000020000000000200000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1, 1), (-1, -1), (2, 1), (-2, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (360360360720)\begin{pmatrix}360 & -360\\ -360 & 720\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=5h8+10h7+15h6+20h5+24h4+16h3+12h2+8h1e=x1g91+x3g78+x2g63+x6g62+x5g60+x4g59e=x10g59+x11g60+x12g62+x8g63+x9g78+x7g91\begin{array}{rcl}h&=&5h_{8}+10h_{7}+15h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ e&=&x_{1} g_{91}+x_{3} g_{78}+x_{2} g_{63}+x_{6} g_{62}+x_{5} g_{60}+x_{4} g_{59}\\ f&=&x_{10} g_{-59}+x_{11} g_{-60}+x_{12} g_{-62}+x_{8} g_{-63}+x_{9} g_{-78}+x_{7} g_{-91}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x7+x3x9+x6x125)h8+(2x1x7+x3x9+x4x10+x5x11+x6x1210)h7+(2x1x7+x2x8+2x3x9+x4x10+2x5x11+x6x1215)h6+(2x1x7+2x2x8+2x3x9+2x4x10+2x5x11+2x6x1220)h5+(3x1x7+3x2x8+2x3x9+2x4x10+2x5x11+2x6x1224)h4+(2x1x7+2x2x8+2x3x9+x4x10+x5x11+x6x1216)h3+(2x1x7+x2x8+x3x9+x4x10+x5x11+x6x1212)h2+(x1x7+x2x8+x3x9+x4x108)h1[e,f] - h = \left(x_{1} x_{7} +x_{3} x_{9} +x_{6} x_{12} -5\right)h_{8}+\left(2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -10\right)h_{7}+\left(2x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -15\right)h_{6}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -20\right)h_{5}+\left(3x_{1} x_{7} +3x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -24\right)h_{4}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -16\right)h_{3}+\left(2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -12\right)h_{2}+\left(x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} -8\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x7+x2x8+x3x9+x4x108=02x1x7+x2x8+x3x9+x4x10+x5x11+x6x1212=02x1x7+2x2x8+2x3x9+x4x10+x5x11+x6x1216=03x1x7+3x2x8+2x3x9+2x4x10+2x5x11+2x6x1224=02x1x7+2x2x8+2x3x9+2x4x10+2x5x11+2x6x1220=02x1x7+x2x8+2x3x9+x4x10+2x5x11+x6x1215=02x1x7+x3x9+x4x10+x5x11+x6x1210=0x1x7+x3x9+x6x125=0\begin{array}{rcl}x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} -8&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -12&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -16&=&0\\3x_{1} x_{7} +3x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -24&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -15&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -10&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{6} x_{12} -5&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=5h8+10h7+15h6+20h5+24h4+16h3+12h2+8h1e=x1g91+x3g78+x2g63+x6g62+x5g60+x4g59f=g59+g60+g62+g63+g78+g91\begin{array}{rcl}h&=&5h_{8}+10h_{7}+15h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\e&=&x_{1} g_{91}+x_{3} g_{78}+x_{2} g_{63}+x_{6} g_{62}+x_{5} g_{60}+x_{4} g_{59}\\f&=&g_{-59}+g_{-60}+g_{-62}+g_{-63}+g_{-78}+g_{-91}\end{array}
Matrix form of the system we are trying to solve: (111100211111222111332222222222212121201111101001)[col. vect.]=(81216242015105)\begin{pmatrix}1 & 1 & 1 & 1 & 0 & 0\\ 2 & 1 & 1 & 1 & 1 & 1\\ 2 & 2 & 2 & 1 & 1 & 1\\ 3 & 3 & 2 & 2 & 2 & 2\\ 2 & 2 & 2 & 2 & 2 & 2\\ 2 & 1 & 2 & 1 & 2 & 1\\ 2 & 0 & 1 & 1 & 1 & 1\\ 1 & 0 & 1 & 0 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 12\\ 16\\ 24\\ 20\\ 15\\ 10\\ 5\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=5h8+10h7+15h6+20h5+24h4+16h3+12h2+8h1e=x1g91+x3g78+x2g63+x6g62+x5g60+x4g59f=x10g59+x11g60+x12g62+x8g63+x9g78+x7g91\begin{array}{rcl}h&=&5h_{8}+10h_{7}+15h_{6}+20h_{5}+24h_{4}+16h_{3}+12h_{2}+8h_{1}\\ e&=&x_{1} g_{91}+x_{3} g_{78}+x_{2} g_{63}+x_{6} g_{62}+x_{5} g_{60}+x_{4} g_{59}\\ f&=&x_{10} g_{-59}+x_{11} g_{-60}+x_{12} g_{-62}+x_{8} g_{-63}+x_{9} g_{-78}+x_{7} g_{-91}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x7+x2x8+x3x9+x4x108=02x1x7+x2x8+x3x9+x4x10+x5x11+x6x1212=02x1x7+2x2x8+2x3x9+x4x10+x5x11+x6x1216=03x1x7+3x2x8+2x3x9+2x4x10+2x5x11+2x6x1224=02x1x7+2x2x8+2x3x9+2x4x10+2x5x11+2x6x1220=02x1x7+x2x8+2x3x9+x4x10+2x5x11+x6x1215=02x1x7+x3x9+x4x10+x5x11+x6x1210=0x1x7+x3x9+x6x125=0\begin{array}{rcl}x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} -8&=&0\\2x_{1} x_{7} +x_{2} x_{8} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -12&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -16&=&0\\3x_{1} x_{7} +3x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -24&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +x_{2} x_{8} +2x_{3} x_{9} +x_{4} x_{10} +2x_{5} x_{11} +x_{6} x_{12} -15&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -10&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{6} x_{12} -5&=&0\\\end{array}

A19A^{9}_1
h-characteristic: (1, 0, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 18
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: 2A2+A12A^{1}_2+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 2V5ψ+10V4ψ+16V3ψ+23V2ψ+18Vψ+17V02V_{5\psi}+10V_{4\psi}+16V_{3\psi}+23V_{2\psi}+18V_{\psi}+17V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=5h8+10h7+14h6+18h5+22h4+15h3+11h2+8h1e=2g98+2g75+g74+2g65+2g49f=g49+g65+g74+g75+g98\begin{array}{rcl}h&=&5h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\\ e&=&2g_{98}+2g_{75}+g_{74}+2g_{65}+2g_{49}\\ f&=&g_{-49}+g_{-65}+g_{-74}+g_{-75}+g_{-98}\end{array}
Lie brackets of the above elements.
h=5h8+10h7+14h6+18h5+22h4+15h3+11h2+8h1e=2g98+2g75+g74+2g65+2g49f=g49+g65+g74+g75+g98\begin{array}{rcl}[e, f]&=&5h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\\ [h, e]&=&4g_{98}+4g_{75}+2g_{74}+4g_{65}+4g_{49}\\ [h, f]&=&-2g_{-49}-2g_{-65}-2g_{-74}-2g_{-75}-2g_{-98}\end{array}
Centralizer type: G2+A13G_2+A^{3}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 17): g12g_{-12}, g5g_{-5}, g4g_{-4}, h4h_{4}, h5h_{5}, h8h3+h2h_{8}-h_{3}+h_{2}, g4g_{4}, g5g_{5}, g12g_{12}, g17+g6g46g_{17}+g_{6}-g_{-46}, g25g13g39g_{25}-g_{13}-g_{-39}, g26g8g27g_{26}-g_{8}-g_{-27}, g27+g8g26g_{27}+g_{-8}-g_{-26}, g31g20+g33g_{31}-g_{20}+g_{-33}, g33g20+g31g_{33}-g_{-20}+g_{-31}, g39+g13g25g_{39}+g_{-13}-g_{-25}, g46g6g17g_{46}-g_{-6}-g_{-17}
Basis of centralizer intersected with cartan (dimension: 3): h5-h_{5}, h4-h_{4}, h8+h3h2-h_{8}+h_{3}-h_{2}
Cartan of centralizer (dimension: 3): h4-h_{4}, h8+h3h2-h_{8}+h_{3}-h_{2}, h5-h_{5}
Cartan-generating semisimple element: 9h8h511h49h3+9h29h_{8}-h_{5}-11h_{4}-9h_{3}+9h_{2}
adjoint action: (12000000000000000009000000000000000002100000000000000000000000000000000000000000000000000000000000000000000000210000000000000000090000000000000000012000000000000000001000000000000000001000000000000000000180000000000000000018000000000000000001100000000000000000110000000000000000010000000000000000001)\begin{pmatrix}12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 21 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -21 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 18 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -11 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
Characteristic polynomial ad H: x171212x15+558774x13125803156x11+14821468929x9879984103032x7+21031135489296x520165847609600x3x^{17}-1212x^{15}+558774x^{13}-125803156x^{11}+14821468929x^9-879984103032x^7+21031135489296x^5-20165847609600x^3
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -21)(x -18)(x -12)(x -11)(x -10)(x -9)(x -1)(x +1)(x +9)(x +10)(x +11)(x +12)(x +18)(x +21)
Eigenvalues of ad H: 00, 2121, 1818, 1212, 1111, 1010, 99, 11, 1-1, 9-9, 10-10, 11-11, 12-12, 18-18, 21-21
17 eigenvectors of ad H: 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)
Centralizer type: G^{1}_2+A^{3}_1
Reductive components (2 total):
Scalar product computed: (130160160190)\begin{pmatrix}1/30 & -1/60\\ -1/60 & 1/90\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
h5h_{5}
matching e: g5g_{5}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (1000000000000000002000000000000000001000000000000000000000000000000000000000000000000000000000000000000000001000000000000000002000000000000000001000000000000000001000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000001)\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
2h5h4-2h_{5}-h_{4}
matching e: g17+g6g46g_{17}+g_{6}-g_{-46}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (3000000000000000003000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003000000000000000003000000000000000002000000000000000001000000000000000000000000000000000000000000000000000001000000000000000001000000000000000001000000000000000002)\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h5h_{5}
matching e: g5g_{5}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (1000000000000000002000000000000000001000000000000000000000000000000000000000000000000000000000000000000000001000000000000000002000000000000000001000000000000000001000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000001)\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
2h5h4-2h_{5}-h_{4}
matching e: g17+g6g46g_{17}+g_{6}-g_{-46}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (3000000000000000003000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003000000000000000003000000000000000002000000000000000001000000000000000000000000000000000000000000000000000001000000000000000001000000000000000001000000000000000002)\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
Elements in Cartan dual to root system: (2, 1), (-2, -1), (1, 1), (-1, -1), (3, 2), (-3, -2), (3, 1), (-3, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120180180360)\begin{pmatrix}120 & -180\\ -180 & 360\\ \end{pmatrix}

Scalar product computed: (190)\begin{pmatrix}1/90\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h8h3+h2h_{8}-h_{3}+h_{2}
matching e: g26g8g27g_{26}-g_{8}-g_{-27}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h8h3+h2h_{8}-h_{3}+h_{2}
matching e: g26g8g27g_{26}-g_{8}-g_{-27}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (360)\begin{pmatrix}360\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=5h8+10h7+14h6+18h5+22h4+15h3+11h2+8h1e=x1g98+x3g75+x5g74+x2g65+x4g49e=x9g49+x7g65+x10g74+x8g75+x6g98\begin{array}{rcl}h&=&5h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\\ e&=&x_{1} g_{98}+x_{3} g_{75}+x_{5} g_{74}+x_{2} g_{65}+x_{4} g_{49}\\ f&=&x_{9} g_{-49}+x_{7} g_{-65}+x_{10} g_{-74}+x_{8} g_{-75}+x_{6} g_{-98}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x6+x4x9+x5x105)h8+(x1x6+x2x7+x3x8+x4x9+2x5x1010)h7+(2x1x6+2x2x7+x3x8+x4x9+2x5x1014)h6+(3x1x6+2x2x7+2x3x8+x4x9+2x5x1018)h5+(4x1x6+2x2x7+3x3x8+x4x9+2x5x1022)h4+(3x1x6+x2x7+2x3x8+x4x9+x5x1015)h3+(2x1x6+x2x7+2x3x8+x5x1011)h2+(x1x6+x2x7+x3x8+x4x98)h1[e,f] - h = \left(x_{1} x_{6} +x_{4} x_{9} +x_{5} x_{10} -5\right)h_{8}+\left(x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -10\right)h_{7}+\left(2x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -14\right)h_{6}+\left(3x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -18\right)h_{5}+\left(4x_{1} x_{6} +2x_{2} x_{7} +3x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -22\right)h_{4}+\left(3x_{1} x_{6} +x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -15\right)h_{3}+\left(2x_{1} x_{6} +x_{2} x_{7} +2x_{3} x_{8} +x_{5} x_{10} -11\right)h_{2}+\left(x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} -8\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x6+x2x7+x3x8+x4x98=02x1x6+x2x7+2x3x8+x5x1011=03x1x6+x2x7+2x3x8+x4x9+x5x1015=04x1x6+2x2x7+3x3x8+x4x9+2x5x1022=03x1x6+2x2x7+2x3x8+x4x9+2x5x1018=02x1x6+2x2x7+x3x8+x4x9+2x5x1014=0x1x6+x2x7+x3x8+x4x9+2x5x1010=0x1x6+x4x9+x5x105=0\begin{array}{rcl}x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} -8&=&0\\2x_{1} x_{6} +x_{2} x_{7} +2x_{3} x_{8} +x_{5} x_{10} -11&=&0\\3x_{1} x_{6} +x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -15&=&0\\4x_{1} x_{6} +2x_{2} x_{7} +3x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -22&=&0\\3x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -18&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -14&=&0\\x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -10&=&0\\x_{1} x_{6} +x_{4} x_{9} +x_{5} x_{10} -5&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=5h8+10h7+14h6+18h5+22h4+15h3+11h2+8h1e=x1g98+x3g75+x5g74+x2g65+x4g49f=g49+g65+g74+g75+g98\begin{array}{rcl}h&=&5h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\\e&=&x_{1} g_{98}+x_{3} g_{75}+x_{5} g_{74}+x_{2} g_{65}+x_{4} g_{49}\\f&=&g_{-49}+g_{-65}+g_{-74}+g_{-75}+g_{-98}\end{array}
Matrix form of the system we are trying to solve: (1111021201312114231232212221121111210011)[col. vect.]=(81115221814105)\begin{pmatrix}1 & 1 & 1 & 1 & 0\\ 2 & 1 & 2 & 0 & 1\\ 3 & 1 & 2 & 1 & 1\\ 4 & 2 & 3 & 1 & 2\\ 3 & 2 & 2 & 1 & 2\\ 2 & 2 & 1 & 1 & 2\\ 1 & 1 & 1 & 1 & 2\\ 1 & 0 & 0 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 11\\ 15\\ 22\\ 18\\ 14\\ 10\\ 5\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=5h8+10h7+14h6+18h5+22h4+15h3+11h2+8h1e=x1g98+x3g75+x5g74+x2g65+x4g49f=x9g49+x7g65+x10g74+x8g75+x6g98\begin{array}{rcl}h&=&5h_{8}+10h_{7}+14h_{6}+18h_{5}+22h_{4}+15h_{3}+11h_{2}+8h_{1}\\ e&=&x_{1} g_{98}+x_{3} g_{75}+x_{5} g_{74}+x_{2} g_{65}+x_{4} g_{49}\\ f&=&x_{9} g_{-49}+x_{7} g_{-65}+x_{10} g_{-74}+x_{8} g_{-75}+x_{6} g_{-98}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x6+x2x7+x3x8+x4x98=02x1x6+x2x7+2x3x8+x5x1011=03x1x6+x2x7+2x3x8+x4x9+x5x1015=04x1x6+2x2x7+3x3x8+x4x9+2x5x1022=03x1x6+2x2x7+2x3x8+x4x9+2x5x1018=02x1x6+2x2x7+x3x8+x4x9+2x5x1014=0x1x6+x2x7+x3x8+x4x9+2x5x1010=0x1x6+x4x9+x5x105=0\begin{array}{rcl}x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} -8&=&0\\2x_{1} x_{6} +x_{2} x_{7} +2x_{3} x_{8} +x_{5} x_{10} -11&=&0\\3x_{1} x_{6} +x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -15&=&0\\4x_{1} x_{6} +2x_{2} x_{7} +3x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -22&=&0\\3x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -18&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -14&=&0\\x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -10&=&0\\x_{1} x_{6} +x_{4} x_{9} +x_{5} x_{10} -5&=&0\\\end{array}

A18A^{8}_1
h-characteristic: (2, 0, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 16
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1: 2A22A^{1}_2 Containing regular semisimple subalgebra number 2: 8A18A^{1}_1 Containing regular semisimple subalgebra number 3: A2+4A1A^{1}_2+4A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 14V4ψ+50V2ψ+28V014V_{4\psi}+50V_{2\psi}+28V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=4h8+8h7+12h6+16h5+20h4+14h3+10h2+8h1e=2g112+2g81+2g49+2g23f=g23+g49+g81+g112\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+8h_{1}\\ e&=&2g_{112}+2g_{81}+2g_{49}+2g_{23}\\ f&=&g_{-23}+g_{-49}+g_{-81}+g_{-112}\end{array}
Lie brackets of the above elements.
h=4h8+8h7+12h6+16h5+20h4+14h3+10h2+8h1e=2g112+2g81+2g49+2g23f=g23+g49+g81+g112\begin{array}{rcl}[e, f]&=&4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+8h_{1}\\ [h, e]&=&4g_{112}+4g_{81}+4g_{49}+4g_{23}\\ [h, f]&=&-2g_{-23}-2g_{-49}-2g_{-81}-2g_{-112}\end{array}
Centralizer type: 2G22G_2
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 28): g14g_{-14}, g11g_{-11}, g7g_{-7}, g6g_{-6}, g4g_{-4}, g3g_{-3}, h3h_{3}, h4h_{4}, h6h_{6}, h7h_{7}, g3g_{3}, g4g_{4}, g6g_{6}, g7g_{7}, g11g_{11}, g14g_{14}, g29+g2+g56g_{29}+g_{2}+g_{-56}, g36g10g50g_{36}-g_{10}-g_{-50}, g42g17+g43g_{42}-g_{-17}+g_{-43}, g43g17+g42g_{43}-g_{17}+g_{-42}, g46g8g74g_{46}-g_{8}-g_{-74}, g50+g10g36g_{50}+g_{-10}-g_{-36}, g54+g15g68g_{54}+g_{15}-g_{-68}, g56+g2+g29g_{56}+g_{-2}+g_{-29}, g60+g22+g62g_{60}+g_{22}+g_{-62}, g62+g22+g60g_{62}+g_{-22}+g_{-60}, g68g15g54g_{68}-g_{-15}-g_{-54}, g74+g8g46g_{74}+g_{-8}-g_{-46}
Basis of centralizer intersected with cartan (dimension: 4): h4-h_{4}, h3-h_{3}, h7-h_{7}, h6-h_{6}
Cartan of centralizer (dimension: 4): h4-h_{4}, h6-h_{6}, h7-h_{7}, h3-h_{3}
Cartan-generating semisimple element: 9h7+7h6h411h39h_{7}+7h_{6}-h_{4}-11h_{3}
adjoint action: (1600000000000000000000000000001200000000000000000000000000001100000000000000000000000000005000000000000000000000000000090000000000000000000000000000210000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002100000000000000000000000000009000000000000000000000000000050000000000000000000000000000110000000000000000000000000000120000000000000000000000000000160000000000000000000000000000100000000000000000000000000001000000000000000000000000000001100000000000000000000000000001100000000000000000000000000009000000000000000000000000000010000000000000000000000000000020000000000000000000000000000100000000000000000000000000007000000000000000000000000000070000000000000000000000000000200000000000000000000000000009)\begin{pmatrix}-16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 21 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -21 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9\\ \end{pmatrix}
Characteristic polynomial ad H: x281424x26+845788x24278973992x22+57045710326x207597970173880x18+670096721894284x1638774892146687768x14+1417688445978452497x1230192821526771954936x10+315571375605764857104x81051963317243181728000x6+765205187054837760000x4x^{28}-1424x^{26}+845788x^{24}-278973992x^{22}+57045710326x^{20}-7597970173880x^{18}+670096721894284x^{16}-38774892146687768x^{14}+1417688445978452497x^{12}-30192821526771954936x^{10}+315571375605764857104x^8-1051963317243181728000x^6+765205187054837760000x^4
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -21)(x -16)(x -12)(x -11)(x -11)(x -10)(x -9)(x -9)(x -7)(x -5)(x -2)(x -1)(x +1)(x +2)(x +5)(x +7)(x +9)(x +9)(x +10)(x +11)(x +11)(x +12)(x +16)(x +21)
Eigenvalues of ad H: 00, 2121, 1616, 1212, 1111, 1010, 99, 77, 55, 22, 11, 1-1, 2-2, 5-5, 7-7, 9-9, 10-10, 11-11, 12-12, 16-16, 21-21
28 eigenvectors of ad H: 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
Centralizer type: 2G^{1}_2
Reductive components (2 total):
Scalar product computed: (130160160190)\begin{pmatrix}1/30 & -1/60\\ -1/60 & 1/90\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
h4h_{4}
matching e: g4g_{4}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000010000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
2h4h3-2h_{4}-h_{3}
matching e: g29+g2+g56g_{29}+g_{2}+g_{-56}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000000000000000000000020000000000000000000000000000100000000000000000000000000001000000000000000000000000000010000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h4h_{4}
matching e: g4g_{4}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000010000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
2h4h3-2h_{4}-h_{3}
matching e: g29+g2+g56g_{29}+g_{2}+g_{-56}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000000000000000000000020000000000000000000000000000100000000000000000000000000001000000000000000000000000000010000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (2, 1), (-2, -1), (1, 1), (-1, -1), (3, 2), (-3, -2), (3, 1), (-3, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120180180360)\begin{pmatrix}120 & -180\\ -180 & 360\\ \end{pmatrix}

Scalar product computed: (130160160190)\begin{pmatrix}1/30 & -1/60\\ -1/60 & 1/90\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
h6h_{6}
matching e: g6g_{6}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (1000000000000000000000000000000000000000000000000000000000100000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000010000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000001000000000000000000000000000010000000000000000000000000000100000000000000000000000000000)\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h7h6h_{7}-h_{6}
matching e: g54+g15g68g_{54}+g_{15}-g_{-68}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000000000000000000000000000000000300000000000000000000000000003000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000000000001000000000000000000000000000010000000000000000000000000000200000000000000000000000000001)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h6h_{6}
matching e: g6g_{6}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (1000000000000000000000000000000000000000000000000000000000100000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000010000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000001000000000000000000000000000010000000000000000000000000000100000000000000000000000000000)\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h7h6h_{7}-h_{6}
matching e: g54+g15g68g_{54}+g_{15}-g_{-68}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (0000000000000000000000000000000000000000000000000000000000300000000000000000000000000003000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003000000000000000000000000000030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000000000001000000000000000000000000000010000000000000000000000000000200000000000000000000000000001)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}
Elements in Cartan dual to root system: (2, 1), (-2, -1), (1, 1), (-1, -1), (3, 2), (-3, -2), (3, 1), (-3, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120180180360)\begin{pmatrix}120 & -180\\ -180 & 360\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=4h8+8h7+12h6+16h5+20h4+14h3+10h2+8h1e=x1g112+x3g81+x2g49+x4g23e=x8g23+x6g49+x7g81+x5g112\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+8h_{1}\\ e&=&x_{1} g_{112}+x_{3} g_{81}+x_{2} g_{49}+x_{4} g_{23}\\ f&=&x_{8} g_{-23}+x_{6} g_{-49}+x_{7} g_{-81}+x_{5} g_{-112}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x5+x2x64)h8+(2x1x5+x2x6+x3x78)h7+(3x1x5+x2x6+2x3x712)h6+(4x1x5+x2x6+3x3x716)h5+(5x1x5+x2x6+3x3x7+x4x820)h4+(3x1x5+x2x6+2x3x7+x4x814)h3+(3x1x5+x3x7+x4x810)h2+(x1x5+x2x6+x3x7+x4x88)h1[e,f] - h = \left(x_{1} x_{5} +x_{2} x_{6} -4\right)h_{8}+\left(2x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} -8\right)h_{7}+\left(3x_{1} x_{5} +x_{2} x_{6} +2x_{3} x_{7} -12\right)h_{6}+\left(4x_{1} x_{5} +x_{2} x_{6} +3x_{3} x_{7} -16\right)h_{5}+\left(5x_{1} x_{5} +x_{2} x_{6} +3x_{3} x_{7} +x_{4} x_{8} -20\right)h_{4}+\left(3x_{1} x_{5} +x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -14\right)h_{3}+\left(3x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -10\right)h_{2}+\left(x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -8\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x5+x2x6+x3x7+x4x88=03x1x5+x3x7+x4x810=03x1x5+x2x6+2x3x7+x4x814=05x1x5+x2x6+3x3x7+x4x820=04x1x5+x2x6+3x3x716=03x1x5+x2x6+2x3x712=02x1x5+x2x6+x3x78=0x1x5+x2x64=0\begin{array}{rcl}x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -8&=&0\\3x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -10&=&0\\3x_{1} x_{5} +x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -14&=&0\\5x_{1} x_{5} +x_{2} x_{6} +3x_{3} x_{7} +x_{4} x_{8} -20&=&0\\4x_{1} x_{5} +x_{2} x_{6} +3x_{3} x_{7} -16&=&0\\3x_{1} x_{5} +x_{2} x_{6} +2x_{3} x_{7} -12&=&0\\2x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} -8&=&0\\x_{1} x_{5} +x_{2} x_{6} -4&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=4h8+8h7+12h6+16h5+20h4+14h3+10h2+8h1e=x1g112+x3g81+x2g49+x4g23f=g23+g49+g81+g112\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+8h_{1}\\e&=&x_{1} g_{112}+x_{3} g_{81}+x_{2} g_{49}+x_{4} g_{23}\\f&=&g_{-23}+g_{-49}+g_{-81}+g_{-112}\end{array}
Matrix form of the system we are trying to solve: (11113011312151314130312021101100)[col. vect.]=(8101420161284)\begin{pmatrix}1 & 1 & 1 & 1\\ 3 & 0 & 1 & 1\\ 3 & 1 & 2 & 1\\ 5 & 1 & 3 & 1\\ 4 & 1 & 3 & 0\\ 3 & 1 & 2 & 0\\ 2 & 1 & 1 & 0\\ 1 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 10\\ 14\\ 20\\ 16\\ 12\\ 8\\ 4\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=4h8+8h7+12h6+16h5+20h4+14h3+10h2+8h1e=x1g112+x3g81+x2g49+x4g23f=x8g23+x6g49+x7g81+x5g112\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+8h_{1}\\ e&=&x_{1} g_{112}+x_{3} g_{81}+x_{2} g_{49}+x_{4} g_{23}\\ f&=&x_{8} g_{-23}+x_{6} g_{-49}+x_{7} g_{-81}+x_{5} g_{-112}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x5+x2x6+x3x7+x4x88=03x1x5+x3x7+x4x810=03x1x5+x2x6+2x3x7+x4x814=05x1x5+x2x6+3x3x7+x4x820=04x1x5+x2x6+3x3x716=03x1x5+x2x6+2x3x712=02x1x5+x2x6+x3x78=0x1x5+x2x64=0\begin{array}{rcl}x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -8&=&0\\3x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -10&=&0\\3x_{1} x_{5} +x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -14&=&0\\5x_{1} x_{5} +x_{2} x_{6} +3x_{3} x_{7} +x_{4} x_{8} -20&=&0\\4x_{1} x_{5} +x_{2} x_{6} +3x_{3} x_{7} -16&=&0\\3x_{1} x_{5} +x_{2} x_{6} +2x_{3} x_{7} -12&=&0\\2x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} -8&=&0\\x_{1} x_{5} +x_{2} x_{6} -4&=&0\\\end{array}

A17A^{7}_1
h-characteristic: (0, 0, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 14
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 7A17A^{1}_1 Containing regular semisimple subalgebra number 2: A2+3A1A^{1}_2+3A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 7V4ψ+14V3ψ+28V2ψ+28Vψ+17V07V_{4\psi}+14V_{3\psi}+28V_{2\psi}+28V_{\psi}+17V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=4h8+8h7+12h6+16h5+20h4+14h3+10h2+7h1e=g106+g91+g83+g72+g71+g70+g69f=g69+g70+g71+g72+g83+g91+g106\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+7h_{1}\\ e&=&g_{106}+g_{91}+g_{83}+g_{72}+g_{71}+g_{70}+g_{69}\\ f&=&g_{-69}+g_{-70}+g_{-71}+g_{-72}+g_{-83}+g_{-91}+g_{-106}\end{array}
Lie brackets of the above elements.
h=4h8+8h7+12h6+16h5+20h4+14h3+10h2+7h1e=g106+g91+g83+g72+g71+g70+g69f=g69+g70+g71+g72+g83+g91+g106\begin{array}{rcl}[e, f]&=&4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+7h_{1}\\ [h, e]&=&2g_{106}+2g_{91}+2g_{83}+2g_{72}+2g_{71}+2g_{70}+2g_{69}\\ [h, f]&=&-2g_{-69}-2g_{-70}-2g_{-71}-2g_{-72}-2g_{-83}-2g_{-91}-2g_{-106}\end{array}
Centralizer type: G22+A1G^{2}_2+A_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 17): g1g_{-1}, h1h_{1}, g1g_{1}, g6g4g4+g6g_{6}-g_{4}-g_{-4}+g_{-6}, g7+g2g2g7g_{7}+g_{2}-g_{-2}-g_{-7}, g8+g4+g4+g8g_{8}+g_{4}+g_{-4}+g_{-8}, g13+g12g12g13g_{13}+g_{12}-g_{-12}-g_{-13}, g14+g10+g10+g14g_{14}+g_{10}+g_{-10}+g_{-14}, g15+g10+g10+g15g_{15}+g_{10}+g_{-10}+g_{-15}, g20+g5+g5+g20g_{20}+g_{5}+g_{-5}+g_{-20}, g21g18+g18g21g_{21}-g_{18}+g_{-18}-g_{-21}, g22+g2g2g22g_{22}+g_{2}-g_{-2}-g_{-22}, g28g26g26+g28g_{28}-g_{26}-g_{-26}+g_{-28}, g29g26g26+g29g_{29}-g_{26}-g_{-26}+g_{-29}, g34g5g5+g34g_{34}-g_{5}-g_{-5}+g_{-34}, g36g18+g18g36g_{36}-g_{18}+g_{-18}-g_{-36}, g42+g12g12g42g_{42}+g_{12}-g_{-12}-g_{-42}
Basis of centralizer intersected with cartan (dimension: 1): h1-h_{1}
Cartan of centralizer (dimension: 3): g1h1g1-g_{1}-h_{1}-g_{-1}, 12791g425891g3612791g3412491g295891g28+2g2617491g2212491g2110791g20+2g1817491g146791g1319491g1217491g109591g8g6+2091g5491g417491g2+17491g2491g4+2091g5g69591g817491g10+19491g12+6791g1317491g142g1810791g20+12491g21+17491g22+2g265891g2812491g2912791g34+5891g36+12791g42-127/91g_{42}-58/91g_{36}-127/91g_{34}-124/91g_{29}-58/91g_{28}+2g_{26}-174/91g_{22}-124/91g_{21}-107/91g_{20}+2g_{18}-174/91g_{14}-67/91g_{13}-194/91g_{12}-174/91g_{10}-95/91g_{8}-g_{6}+20/91g_{5}-4/91g_{4}-174/91g_{2}+174/91g_{-2}-4/91g_{-4}+20/91g_{-5}-g_{-6}-95/91g_{-8}-174/91g_{-10}+194/91g_{-12}+67/91g_{-13}-174/91g_{-14}-2g_{-18}-107/91g_{-20}+124/91g_{-21}+174/91g_{-22}+2g_{-26}-58/91g_{-28}-124/91g_{-29}-127/91g_{-34}+58/91g_{-36}+127/91g_{-42}, 3691g42+3391g363691g343391g29+3391g288391g223391g211691g20+g158391g14+2491g131291g12+891g10491g8+g7+2091g5491g4+891g2891g2491g4+2091g5g7491g8+891g10+1291g122491g138391g14+g151691g20+3391g21+8391g22+3391g283391g293691g343391g36+3691g42-36/91g_{42}+33/91g_{36}-36/91g_{34}-33/91g_{29}+33/91g_{28}-83/91g_{22}-33/91g_{21}-16/91g_{20}+g_{15}-83/91g_{14}+24/91g_{13}-12/91g_{12}+8/91g_{10}-4/91g_{8}+g_{7}+20/91g_{5}-4/91g_{4}+8/91g_{2}-8/91g_{-2}-4/91g_{-4}+20/91g_{-5}-g_{-7}-4/91g_{-8}+8/91g_{-10}+12/91g_{-12}-24/91g_{-13}-83/91g_{-14}+g_{-15}-16/91g_{-20}+33/91g_{-21}+83/91g_{-22}+33/91g_{-28}-33/91g_{-29}-36/91g_{-34}-33/91g_{-36}+36/91g_{-42}
Cartan-generating semisimple element: g42+g36+g34+g29+g282g26+g22+g21+g202g18+g15+g14+g13+2g12+2g10+g8+g7+g6+2g2+g1+h1+g12g2+g6g7+g8+2g102g12g13+g14+g15+2g18+g20g21g222g26+g28+g29+g34g36g42g_{42}+g_{36}+g_{34}+g_{29}+g_{28}-2g_{26}+g_{22}+g_{21}+g_{20}-2g_{18}+g_{15}+g_{14}+g_{13}+2g_{12}+2g_{10}+g_{8}+g_{7}+g_{6}+2g_{2}+g_{1}+h_{1}+g_{-1}-2g_{-2}+g_{-6}-g_{-7}+g_{-8}+2g_{-10}-2g_{-12}-g_{-13}+g_{-14}+g_{-15}+2g_{-18}+g_{-20}-g_{-21}-g_{-22}-2g_{-26}+g_{-28}+g_{-29}+g_{-34}-g_{-36}-g_{-42}
adjoint action: (2200000000000000010100000000000000022000000000000000000101113212212100010111111021200000010011011111110001110231102310000011110011120311000111100101100120003311010023201100011011110001300000101111110001330001201201010021100010011011000102000031031031310110001101011031000300001101301313130)\begin{pmatrix}-2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 1 & 1 & 1 & -3 & 2 & -1 & -2 & -2 & 1 & 2 & 1\\ 0 & 0 & 0 & -1 & 0 & 1 & 1 & 1 & -1 & -1 & -1 & 0 & 2 & 1 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & -1 & 0 & 1 & 1 & -1 & -1 & -1 & 1 & 1\\ 0 & 0 & 0 & 1 & -1 & -1 & 0 & 2 & 3 & -1 & 1 & 0 & -2 & -3 & 1 & 0 & 0\\ 0 & 0 & 0 & -1 & 1 & 1 & 1 & 0 & 0 & -1 & -1 & -1 & -2 & 0 & 3 & -1 & 1\\ 0 & 0 & 0 & -1 & -1 & 1 & 1 & 0 & 0 & -1 & 0 & 1 & 1 & 0 & 0 & 1 & -2\\ 0 & 0 & 0 & 3 & -3 & -1 & -1 & 0 & 1 & 0 & 0 & -2 & 3 & 2 & 0 & -1 & -1\\ 0 & 0 & 0 & 1 & 1 & 0 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & -1 & 3 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 1 & 1 & -1 & 1 & -1 & -1 & 0 & 0 & 0 & 1 & -3 & 3\\ 0 & 0 & 0 & 1 & 2 & 0 & -1 & 2 & 0 & -1 & 0 & 1 & 0 & 0 & -2 & -1 & -1\\ 0 & 0 & 0 & 1 & 0 & 0 & -1 & -1 & 0 & -1 & -1 & 0 & 0 & 0 & 1 & 0 & 2\\ 0 & 0 & 0 & 0 & -3 & 1 & 0 & -3 & -1 & 0 & 3 & -1 & 3 & 1 & 0 & 1 & -1\\ 0 & 0 & 0 & 1 & 1 & 0 & -1 & 0 & 1 & -1 & 0 & 3 & -1 & 0 & 0 & 0 & -3\\ 0 & 0 & 0 & 0 & -1 & 1 & 0 & -1 & -3 & 0 & 1 & -3 & 1 & 3 & -1 & 3 & 0\\ \end{pmatrix}
Characteristic polynomial ad H: x1742x1344x11847x91596x71056x5+3584x3x^{17}-42x^{13}-44x^{11}-847x^9-1596x^7-1056x^5+3584x^3
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -1)(x +1)(x^2-8)(x^2+7)(x^2-3x +4)(x^2-x +2)(x^2+x +2)(x^2+3x +4)
Eigenvalues of ad H: 00, 11, 1-1, 222\sqrt{2}, 22-2\sqrt{2}, 7\sqrt{-7}, 7-\sqrt{-7}, 127+321/2\sqrt{-7}+3/2, 127+32-1/2\sqrt{-7}+3/2, 127+121/2\sqrt{-7}+1/2, 127+12-1/2\sqrt{-7}+1/2, 127121/2\sqrt{-7}-1/2, 12712-1/2\sqrt{-7}-1/2, 127321/2\sqrt{-7}-3/2, 12732-1/2\sqrt{-7}-3/2
17 eigenvectors of ad H: 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 12/23, 127/69, 32/69, 20/23, -47/69, 127/69, 20/69, 1/23, -47/69, 1, 1/23, 0, 1, 0(0,0,0,1223,12769,3269,2023,4769,12769,2069,123,4769,1,123,0,1,0), 0, 0, 0, 11/23, -58/69, 37/69, 3/23, 116/69, -58/69, 49/69, 22/23, 116/69, 0, 22/23, 1, 0, 1(0,0,0,1123,5869,3769,323,11669,5869,4969,2223,11669,0,2223,1,0,1), 0, 0, 0, 5/48, -45/32, 5/24, -5/8, 31/24, -143/96, 17/48, 49/48, 11/8, -67/96, 49/48, 23/24, -67/96, 1(0,0,0,548,4532,524,58,3124,14396,1748,4948,118,6796,4948,2324,6796,1), 0, 0, 0, 0, -3, 0, -1/2, 3, -3, 1/2, 0, 3, 0, 0, 1, 0, 1(0,0,0,0,3,0,12,3,3,12,0,3,0,0,1,0,1), 2\sqrt{2}-3, -\sqrt{2}+1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(223,2+1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), -2\sqrt{2}-3, \sqrt{2}+1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(223,2+1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 1/8\sqrt{-7}-1/8, 11/16\sqrt{-7}-9/16, 0, 1/16\sqrt{-7}-7/16, -11/16\sqrt{-7}+1/16, 11/16\sqrt{-7}-9/16, 1/16\sqrt{-7}+13/16, -7/32\sqrt{-7}+29/32, -11/16\sqrt{-7}+1/16, 15/32\sqrt{-7}-37/32, -7/32\sqrt{-7}+29/32, 1, 15/32\sqrt{-7}-37/32, 1(0,0,0,18718,11167916,0,1167716,11167+116,11167916,1167+1316,7327+2932,11167+116,153273732,7327+2932,1,153273732,1), 0, 0, 0, -1/8\sqrt{-7}-1/8, -11/16\sqrt{-7}-9/16, 0, -1/16\sqrt{-7}-7/16, 11/16\sqrt{-7}+1/16, -11/16\sqrt{-7}-9/16, -1/16\sqrt{-7}+13/16, 7/32\sqrt{-7}+29/32, 11/16\sqrt{-7}+1/16, -15/32\sqrt{-7}-37/32, 7/32\sqrt{-7}+29/32, 1, -15/32\sqrt{-7}-37/32, 1(0,0,0,18718,11167916,0,1167716,11167+116,11167916,1167+1316,7327+2932,11167+116,153273732,7327+2932,1,153273732,1), 0, 0, 0, -1595/20248\sqrt{-7}+4031/20248, 7253/20248\sqrt{-7}-20079/20248, 143/5062\sqrt{-7}+651/5062, -3119/20248\sqrt{-7}-11261/20248, -5603/20248\sqrt{-7}+15909/20248, 5733/20248\sqrt{-7}-19919/20248, -1367/10124\sqrt{-7}+4007/10124, -525/20248\sqrt{-7}+22035/20248, -4303/20248\sqrt{-7}+16305/20248, 4579/20248\sqrt{-7}-13137/20248, -169/20248\sqrt{-7}+20399/20248, -55/2531\sqrt{-7}+2670/2531, 3835/20248\sqrt{-7}-11993/20248, 1(0,0,0,1595202487+403120248,72532024872007920248,14350627+6515062,31192024871126120248,5603202487+1590920248,57332024871991920248,1367101247+400710124,525202487+2203520248,4303202487+1630520248,45792024871313720248,169202487+2039920248,5525317+26702531,38352024871199320248,1), 0, 0, 0, 1595/20248\sqrt{-7}+4031/20248, -7253/20248\sqrt{-7}-20079/20248, -143/5062\sqrt{-7}+651/5062, 3119/20248\sqrt{-7}-11261/20248, 5603/20248\sqrt{-7}+15909/20248, -5733/20248\sqrt{-7}-19919/20248, 1367/10124\sqrt{-7}+4007/10124, 525/20248\sqrt{-7}+22035/20248, 4303/20248\sqrt{-7}+16305/20248, -4579/20248\sqrt{-7}-13137/20248, 169/20248\sqrt{-7}+20399/20248, 55/2531\sqrt{-7}+2670/2531, -3835/20248\sqrt{-7}-11993/20248, 1(0,0,0,1595202487+403120248,72532024872007920248,14350627+6515062,31192024871126120248,5603202487+1590920248,57332024871991920248,1367101247+400710124,525202487+2203520248,4303202487+1630520248,45792024871313720248,169202487+2039920248,5525317+26702531,38352024871199320248,1), 0, 0, 0, -11/1061\sqrt{-7}+127/1061, 2017/4244\sqrt{-7}-6697/4244, -11/2122\sqrt{-7}+127/2122, -207/4244\sqrt{-7}-2047/4244, -1907/4244\sqrt{-7}+5427/4244, 487/1061\sqrt{-7}-1668/1061, -207/4244\sqrt{-7}+2197/4244, -95/4244\sqrt{-7}+4955/4244, -919/2122\sqrt{-7}+2701/2122, 741/4244\sqrt{-7}-4697/4244, -21/1061\sqrt{-7}+1207/1061, 1, 365/2122\sqrt{-7}-2285/2122, 1(0,0,0,1110617+1271061,20174244766974244,1121227+1272122,2074244720474244,190742447+54274244,4871061716681061,20742447+21974244,9542447+49554244,91921227+27012122,7414244746974244,2110617+12071061,1,3652122722852122,1), 0, 0, 0, 11/1061\sqrt{-7}+127/1061, -2017/4244\sqrt{-7}-6697/4244, 11/2122\sqrt{-7}+127/2122, 207/4244\sqrt{-7}-2047/4244, 1907/4244\sqrt{-7}+5427/4244, -487/1061\sqrt{-7}-1668/1061, 207/4244\sqrt{-7}+2197/4244, 95/4244\sqrt{-7}+4955/4244, 919/2122\sqrt{-7}+2701/2122, -741/4244\sqrt{-7}-4697/4244, 21/1061\sqrt{-7}+1207/1061, 1, -365/2122\sqrt{-7}-2285/2122, 1(0,0,0,1110617+1271061,20174244766974244,1121227+1272122,2074244720474244,190742447+54274244,4871061716681061,20742447+21974244,9542447+49554244,91921227+27012122,7414244746974244,2110617+12071061,1,3652122722852122,1), 0, 0, 0, 0, 15/16\sqrt{-7}-41/16, 0, -1/2, -15/16\sqrt{-7}+37/16, 15/16\sqrt{-7}-41/16, 1/2, 1/16\sqrt{-7}+29/16, -15/16\sqrt{-7}+37/16, 1/16\sqrt{-7}-31/16, 1/16\sqrt{-7}+29/16, 1, 1/16\sqrt{-7}-31/16, 1(0,0,0,0,151674116,0,12,15167+3716,151674116,12,1167+2916,15167+3716,11673116,1167+2916,1,11673116,1), 0, 0, 0, 0, -15/16\sqrt{-7}-41/16, 0, -1/2, 15/16\sqrt{-7}+37/16, -15/16\sqrt{-7}-41/16, 1/2, -1/16\sqrt{-7}+29/16, 15/16\sqrt{-7}+37/16, -1/16\sqrt{-7}-31/16, -1/16\sqrt{-7}+29/16, 1, -1/16\sqrt{-7}-31/16, 1(0,0,0,0,151674116,0,12,15167+3716,151674116,12,1167+2916,15167+3716,11673116,1167+2916,1,11673116,1), 0, 0, 0, 0, -1/2\sqrt{-7}+1/2, 0, 0, 1/2\sqrt{-7}-1/2, -1/2\sqrt{-7}+1/2, 0, -1, 1/2\sqrt{-7}-1/2, 1, -1, 0, 1, 0(0,0,0,0,127+12,0,0,12712,127+12,0,1,12712,1,1,0,1,0), 0, 0, 0, 0, 1/2\sqrt{-7}+1/2, 0, 0, -1/2\sqrt{-7}-1/2, 1/2\sqrt{-7}+1/2, 0, -1, -1/2\sqrt{-7}-1/2, 1, -1, 0, 1, 0(0,0,0,0,127+12,0,0,12712,127+12,0,1,12712,1,1,0,1,0)
Centralizer type: G^{2}_2+A^{1}_1
Reductive components (2 total):
Scalar product computed: (160112011201180)\begin{pmatrix}1/60 & -1/120\\ -1/120 & 1/180\\ \end{pmatrix}
Simple basis of Cartan of centralizer (2 total):
877g4237287g36+877g34+29287g2937287g28+277g26+79287g22+29287g21+377g20+277g1895287g15+79287g147g13+177g12477g1095287g7177g6577g5+177g4477g2+477g2+177g4577g5177g6+95287g7477g10177g12+7g13+79287g1495287g15277g18+377g2029287g2179287g22+277g2637287g28+29287g29+877g34+37287g36877g428/7\sqrt{-7}g_{42}-37/28\sqrt{-7}g_{36}+8/7\sqrt{-7}g_{34}+29/28\sqrt{-7}g_{29}-37/28\sqrt{-7}g_{28}+2/7\sqrt{-7}g_{26}+79/28\sqrt{-7}g_{22}+29/28\sqrt{-7}g_{21}+3/7\sqrt{-7}g_{20}+2/7\sqrt{-7}g_{18}-95/28\sqrt{-7}g_{15}+79/28\sqrt{-7}g_{14}-\sqrt{-7}g_{13}+1/7\sqrt{-7}g_{12}-4/7\sqrt{-7}g_{10}-95/28\sqrt{-7}g_{7}-1/7\sqrt{-7}g_{6}-5/7\sqrt{-7}g_{5}+1/7\sqrt{-7}g_{4}-4/7\sqrt{-7}g_{2}+4/7\sqrt{-7}g_{-2}+1/7\sqrt{-7}g_{-4}-5/7\sqrt{-7}g_{-5}-1/7\sqrt{-7}g_{-6}+95/28\sqrt{-7}g_{-7}-4/7\sqrt{-7}g_{-10}-1/7\sqrt{-7}g_{-12}+\sqrt{-7}g_{-13}+79/28\sqrt{-7}g_{-14}-95/28\sqrt{-7}g_{-15}-2/7\sqrt{-7}g_{-18}+3/7\sqrt{-7}g_{-20}-29/28\sqrt{-7}g_{-21}-79/28\sqrt{-7}g_{-22}+2/7\sqrt{-7}g_{-26}-37/28\sqrt{-7}g_{-28}+29/28\sqrt{-7}g_{-29}+8/7\sqrt{-7}g_{-34}+37/28\sqrt{-7}g_{-36}-8/7\sqrt{-7}g_{-42}
matching e: g42+(153273732)g36+g34+(7327+2932)g29+(153273732)g28+(147+14)g26+(11167+116)g22+(7327+2932)g21+(1167+1316)g20+(147+14)g18+(11167916)g15+(11167+116)g14+(1167716)g13+(1167+916)g1212g10+(11167916)g7+(18718)g6+(1167316)g5+(187+18)g412g2+12g2+(187+18)g4+(1167316)g5+(18718)g6+(11167+916)g712g10+(1167916)g12+(1167+716)g13+(11167+116)g14+(11167916)g15+(14714)g18+(1167+1316)g20+(73272932)g21+(11167116)g22+(147+14)g26+(153273732)g28+(7327+2932)g29+g34+(15327+3732)g36g42g_{42}+\left(15/32\sqrt{-7}-37/32\right)g_{36}+g_{34}+\left(-7/32\sqrt{-7}+29/32\right)g_{29}+\left(15/32\sqrt{-7}-37/32\right)g_{28}+\left(-1/4\sqrt{-7}+1/4\right)g_{26}+\left(-11/16\sqrt{-7}+1/16\right)g_{22}+\left(-7/32\sqrt{-7}+29/32\right)g_{21}+\left(1/16\sqrt{-7}+13/16\right)g_{20}+\left(-1/4\sqrt{-7}+1/4\right)g_{18}+\left(11/16\sqrt{-7}-9/16\right)g_{15}+\left(-11/16\sqrt{-7}+1/16\right)g_{14}+\left(1/16\sqrt{-7}-7/16\right)g_{13}+\left(1/16\sqrt{-7}+9/16\right)g_{12}-1/2g_{10}+\left(11/16\sqrt{-7}-9/16\right)g_{7}+\left(1/8\sqrt{-7}-1/8\right)g_{6}+\left(1/16\sqrt{-7}-3/16\right)g_{5}+\left(-1/8\sqrt{-7}+1/8\right)g_{4}-1/2g_{2}+1/2g_{-2}+\left(-1/8\sqrt{-7}+1/8\right)g_{-4}+\left(1/16\sqrt{-7}-3/16\right)g_{-5}+\left(1/8\sqrt{-7}-1/8\right)g_{-6}+\left(-11/16\sqrt{-7}+9/16\right)g_{-7}-1/2g_{-10}+\left(-1/16\sqrt{-7}-9/16\right)g_{-12}+\left(-1/16\sqrt{-7}+7/16\right)g_{-13}+\left(-11/16\sqrt{-7}+1/16\right)g_{-14}+\left(11/16\sqrt{-7}-9/16\right)g_{-15}+\left(1/4\sqrt{-7}-1/4\right)g_{-18}+\left(1/16\sqrt{-7}+13/16\right)g_{-20}+\left(7/32\sqrt{-7}-29/32\right)g_{-21}+\left(11/16\sqrt{-7}-1/16\right)g_{-22}+\left(-1/4\sqrt{-7}+1/4\right)g_{-26}+\left(15/32\sqrt{-7}-37/32\right)g_{-28}+\left(-7/32\sqrt{-7}+29/32\right)g_{-29}+g_{-34}+\left(-15/32\sqrt{-7}+37/32\right)g_{-36}-g_{-42}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action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begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -79/28\sqrt{-7} & 0 & -2/7\sqrt{-7} & -95/28\sqrt{-7} & 79/28\sqrt{-7} & 6/7\sqrt{-7} & -2/7\sqrt{-7} & 95/28\sqrt{-7} & 2/7\sqrt{-7} & 2/7\sqrt{-7} & -\sqrt{-7} & -2/7\sqrt{-7} & 3/7\sqrt{-7}\\ 0 & 0 & 0 & -79/28\sqrt{-7} & 0 & -95/28\sqrt{-7} & 29/28\sqrt{-7} & -1/7\sqrt{-7} & 0 & 37/28\sqrt{-7} & \sqrt{-7} & 0 & 11/7\sqrt{-7} & 8/7\sqrt{-7} & 2/7\sqrt{-7} & 0 & 0\\ 0 & 0 & 0 & 0 & -95/28\sqrt{-7} & 0 & 0 & -79/28\sqrt{-7} & 95/28\sqrt{-7} & 0 & 29/28\sqrt{-7} & 79/28\sqrt{-7} & 37/28\sqrt{-7} & -29/28\sqrt{-7} & -8/7\sqrt{-7} & -37/28\sqrt{-7} & 8/7\sqrt{-7}\\ 0 & 0 & 0 & -2/7\sqrt{-7} & -29/28\sqrt{-7} & -3/7\sqrt{-7} & 0 & -2/7\sqrt{-7} & 3/4\sqrt{-7} & 2/7\sqrt{-7} & -95/28\sqrt{-7} & 0 & 4/7\sqrt{-7} & 111/28\sqrt{-7} & -1/7\sqrt{-7} & 0 & 0\\ 0 & 0 & 0 & 95/28\sqrt{-7} & -1/7\sqrt{-7} & 79/28\sqrt{-7} & -37/28\sqrt{-7} & 0 & 0 & -29/28\sqrt{-7} & -13/7\sqrt{-7} & 0 & -1/7\sqrt{-7} & 0 & 3/4\sqrt{-7} & -8/7\sqrt{-7} & -37/28\sqrt{-7}\\ 0 & 0 & 0 & -79/28\sqrt{-7} & 0 & -95/28\sqrt{-7} & 29/28\sqrt{-7} & 0 & 0 & 37/28\sqrt{-7} & 0 & -1/7\sqrt{-7} & 8/7\sqrt{-7} & 15/7\sqrt{-7} & 0 & 3/7\sqrt{-7} & 2/7\sqrt{-7}\\ 0 & 0 & 0 & -6/7\sqrt{-7} & 45/28\sqrt{-7} & \sqrt{-7} & 2/7\sqrt{-7} & 0 & -37/28\sqrt{-7} & 0 & 0 & 2/7\sqrt{-7} & -111/28\sqrt{-7} & -4/7\sqrt{-7} & 0 & 95/28\sqrt{-7} & 1/7\sqrt{-7}\\ 0 & 0 & 0 & -37/28\sqrt{-7} & -\sqrt{-7} & 33/14\sqrt{-7} & 95/28\sqrt{-7} & -13/7\sqrt{-7} & -8/7\sqrt{-7} & -79/28\sqrt{-7} & 0 & 0 & 1/7\sqrt{-7} & 0 & 9/4\sqrt{-7} & 0 & 0\\ 0 & 0 & 0 & 95/28\sqrt{-7} & 0 & 79/28\sqrt{-7} & -37/28\sqrt{-7} & 0 & -1/7\sqrt{-7} & -29/28\sqrt{-7} & -8/7\sqrt{-7} & 0 & 0 & -5/7\sqrt{-7} & 29/28\sqrt{-7} & -9/7\sqrt{-7} & -45/28\sqrt{-7}\\ 0 & 0 & 0 & 29/28\sqrt{-7} & 11/7\sqrt{-7} & -33/14\sqrt{-7} & -79/28\sqrt{-7} & 1/7\sqrt{-7} & 0 & 95/28\sqrt{-7} & 1/7\sqrt{-7} & 8/7\sqrt{-7} & 0 & 0 & 4/7\sqrt{-7} & 0 & -79/28\sqrt{-7}\\ 0 & 0 & 0 & -37/28\sqrt{-7} & 0 & 33/14\sqrt{-7} & 95/28\sqrt{-7} & -8/7\sqrt{-7} & -15/7\sqrt{-7} & -79/28\sqrt{-7} & 0 & -5/7\sqrt{-7} & 0 & 0 & 79/28\sqrt{-7} & 1/7\sqrt{-7} & -4/7\sqrt{-7}\\ 0 & 0 & 0 & 0 & 45/28\sqrt{-7} & 8/7\sqrt{-7} & 0 & -3/4\sqrt{-7} & -29/28\sqrt{-7} & 0 & 9/4\sqrt{-7} & 37/28\sqrt{-7} & -111/28\sqrt{-7} & -95/28\sqrt{-7} & 0 & 79/28\sqrt{-7} & 0\\ 0 & 0 & 0 & 29/28\sqrt{-7} & 8/7\sqrt{-7} & -33/14\sqrt{-7} & -79/28\sqrt{-7} & 0 & 3/7\sqrt{-7} & 95/28\sqrt{-7} & 0 & 9/7\sqrt{-7} & 0 & 1/7\sqrt{-7} & 0 & 0 & -9/4\sqrt{-7}\\ 0 & 0 & 0 & 0 & 37/28\sqrt{-7} & 8/7\sqrt{-7} & 0 & -29/28\sqrt{-7} & -3/4\sqrt{-7} & 0 & 79/28\sqrt{-7} & 45/28\sqrt{-7} & -95/28\sqrt{-7} & -111/28\sqrt{-7} & 0 & 9/4\sqrt{-7} & 0\\ \end{pmatrix}
(1277+5)g42+(111567298)g36+(1277+5)g34+(87567+378)g29+(111567298)g28+(3771)g26+(237567+878)g22+(87567+378)g21+(9147+52)g20+(3771)g18+(285567878)g15+(237567+878)g14+(32752)g13+(3147+52)g12+677g10+g8+(285567878)g7+(3147+12)g6+(1514752)g5+(3147+12)g4+677g2677g2+(3147+12)g4+(1514752)g5+(3147+12)g6+(285567+878)g7+g8+677g10+(314752)g12+(327+52)g13+(237567+878)g14+(285567878)g15+(377+1)g18+(9147+52)g20+(87567378)g21+(237567878)g22+(3771)g26+(111567298)g28+(87567+378)g29+(1277+5)g34+(111567+298)g36+(12775)g42\left(-12/7\sqrt{-7}+5\right)g_{42}+\left(111/56\sqrt{-7}-29/8\right)g_{36}+\left(-12/7\sqrt{-7}+5\right)g_{34}+\left(-87/56\sqrt{-7}+37/8\right)g_{29}+\left(111/56\sqrt{-7}-29/8\right)g_{28}+\left(-3/7\sqrt{-7}-1\right)g_{26}+\left(-237/56\sqrt{-7}+87/8\right)g_{22}+\left(-87/56\sqrt{-7}+37/8\right)g_{21}+\left(-9/14\sqrt{-7}+5/2\right)g_{20}+\left(-3/7\sqrt{-7}-1\right)g_{18}+\left(285/56\sqrt{-7}-87/8\right)g_{15}+\left(-237/56\sqrt{-7}+87/8\right)g_{14}+\left(3/2\sqrt{-7}-5/2\right)g_{13}+\left(-3/14\sqrt{-7}+5/2\right)g_{12}+6/7\sqrt{-7}g_{10}+g_{8}+\left(285/56\sqrt{-7}-87/8\right)g_{7}+\left(3/14\sqrt{-7}+1/2\right)g_{6}+\left(15/14\sqrt{-7}-5/2\right)g_{5}+\left(-3/14\sqrt{-7}+1/2\right)g_{4}+6/7\sqrt{-7}g_{2}-6/7\sqrt{-7}g_{-2}+\left(-3/14\sqrt{-7}+1/2\right)g_{-4}+\left(15/14\sqrt{-7}-5/2\right)g_{-5}+\left(3/14\sqrt{-7}+1/2\right)g_{-6}+\left(-285/56\sqrt{-7}+87/8\right)g_{-7}+g_{-8}+6/7\sqrt{-7}g_{-10}+\left(3/14\sqrt{-7}-5/2\right)g_{-12}+\left(-3/2\sqrt{-7}+5/2\right)g_{-13}+\left(-237/56\sqrt{-7}+87/8\right)g_{-14}+\left(285/56\sqrt{-7}-87/8\right)g_{-15}+\left(3/7\sqrt{-7}+1\right)g_{-18}+\left(-9/14\sqrt{-7}+5/2\right)g_{-20}+\left(87/56\sqrt{-7}-37/8\right)g_{-21}+\left(237/56\sqrt{-7}-87/8\right)g_{-22}+\left(-3/7\sqrt{-7}-1\right)g_{-26}+\left(111/56\sqrt{-7}-29/8\right)g_{-28}+\left(-87/56\sqrt{-7}+37/8\right)g_{-29}+\left(-12/7\sqrt{-7}+5\right)g_{-34}+\left(-111/56\sqrt{-7}+29/8\right)g_{-36}+\left(12/7\sqrt{-7}-5\right)g_{-42}
matching e: g42+(3652122722852122)g36+g34+(2110617+12071061)g29+(7414244746974244)g28+(657424471314244)g26+(91921227+27012122)g22+(9542447+49554244)g21+(20742447+21974244)g20+(635424473854244)g18+(4871061716681061)g15+(190742447+54274244)g14+(2074244720474244)g13+(20742447+21974244)g12+(414244712454244)g10+(1121227+1272122)g8+(20174244766974244)g7+(1110617+1271061)g6+(2074244720474244)g5+(11212271272122)g4+(1794244712954244)g2+(17942447+12954244)g2+(11212271272122)g4+(2074244720474244)g5+(1110617+1271061)g6+(201742447+66974244)g7+(1121227+1272122)g8+(414244712454244)g10+(2074244721974244)g12+(20742447+20474244)g13+(190742447+54274244)g14+(4871061716681061)g15+(63542447+3854244)g18+(20742447+21974244)g20+(954244749554244)g21+(9192122727012122)g22+(657424471314244)g26+(7414244746974244)g28+(2110617+12071061)g29+g34+(36521227+22852122)g36g42g_{42}+\left(-365/2122\sqrt{-7}-2285/2122\right)g_{36}+g_{34}+\left(21/1061\sqrt{-7}+1207/1061\right)g_{29}+\left(-741/4244\sqrt{-7}-4697/4244\right)g_{28}+\left(657/4244\sqrt{-7}-131/4244\right)g_{26}+\left(919/2122\sqrt{-7}+2701/2122\right)g_{22}+\left(95/4244\sqrt{-7}+4955/4244\right)g_{21}+\left(207/4244\sqrt{-7}+2197/4244\right)g_{20}+\left(635/4244\sqrt{-7}-385/4244\right)g_{18}+\left(-487/1061\sqrt{-7}-1668/1061\right)g_{15}+\left(1907/4244\sqrt{-7}+5427/4244\right)g_{14}+\left(207/4244\sqrt{-7}-2047/4244\right)g_{13}+\left(207/4244\sqrt{-7}+2197/4244\right)g_{12}+\left(-41/4244\sqrt{-7}-1245/4244\right)g_{10}+\left(11/2122\sqrt{-7}+127/2122\right)g_{8}+\left(-2017/4244\sqrt{-7}-6697/4244\right)g_{7}+\left(11/1061\sqrt{-7}+127/1061\right)g_{6}+\left(207/4244\sqrt{-7}-2047/4244\right)g_{5}+\left(-11/2122\sqrt{-7}-127/2122\right)g_{4}+\left(-179/4244\sqrt{-7}-1295/4244\right)g_{2}+\left(179/4244\sqrt{-7}+1295/4244\right)g_{-2}+\left(-11/2122\sqrt{-7}-127/2122\right)g_{-4}+\left(207/4244\sqrt{-7}-2047/4244\right)g_{-5}+\left(11/1061\sqrt{-7}+127/1061\right)g_{-6}+\left(2017/4244\sqrt{-7}+6697/4244\right)g_{-7}+\left(11/2122\sqrt{-7}+127/2122\right)g_{-8}+\left(-41/4244\sqrt{-7}-1245/4244\right)g_{-10}+\left(-207/4244\sqrt{-7}-2197/4244\right)g_{-12}+\left(-207/4244\sqrt{-7}+2047/4244\right)g_{-13}+\left(1907/4244\sqrt{-7}+5427/4244\right)g_{-14}+\left(-487/1061\sqrt{-7}-1668/1061\right)g_{-15}+\left(-635/4244\sqrt{-7}+385/4244\right)g_{-18}+\left(207/4244\sqrt{-7}+2197/4244\right)g_{-20}+\left(-95/4244\sqrt{-7}-4955/4244\right)g_{-21}+\left(-919/2122\sqrt{-7}-2701/2122\right)g_{-22}+\left(657/4244\sqrt{-7}-131/4244\right)g_{-26}+\left(-741/4244\sqrt{-7}-4697/4244\right)g_{-28}+\left(21/1061\sqrt{-7}+1207/1061\right)g_{-29}+g_{-34}+\left(365/2122\sqrt{-7}+2285/2122\right)g_{-36}-g_{-42}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (00000000000000000000000000000000000000000000000000000002375678780377285567878237567+878977377+1285567+8783771377132752377+19147+52000237567878028556787887567+3783147+121111567+298327+52033147+1521277+5377100000028556787800237567878285567+878087567+378237567+878111567+29887567378127751115672981277+5000377875673789147520377+1987+4583772855678780677333567+8783147+1200000285567+8783147+12237567+87811156729800875673783914715213147520987+45812775111567298000237567878128556787887567+37800111567+29803147+121277+545147+15209147+523771000977135567+218327+5237701115672980037713335678786770285567+8783147120001115672983275299287+334285567+8783914715212775237567878003147+1212787+87800000285567+8780237567+87811156729813147+12875673781277500151475287567+3782714715213556721800087567+37833147+152992873342375678783147+520285567+8783147+121277+5006771237567878000111567298099287+334285567+87812775451471522375678781151475200237567+8783147+126770000135567+2181277+509874588756737802787+878111567+2983335678782855678780237567+878100087567+3781277+59928733423756787809147+52285567+878027147+15213147+120027878780000111567+2981277+50875673789874580237567+878135567+21828556787833356787812787+8780)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 237/56\sqrt{-7}-87/8 & 0 & 3/7\sqrt{-7} & 285/56\sqrt{-7}-87/8 & -237/56\sqrt{-7}+87/8 & -9/7\sqrt{-7} & 3/7\sqrt{-7}+1 & -285/56\sqrt{-7}+87/8 & -3/7\sqrt{-7}-1 & -3/7\sqrt{-7}-1 & 3/2\sqrt{-7}-5/2 & 3/7\sqrt{-7}+1 & -9/14\sqrt{-7}+5/2\\ 0 & 0 & 0 & 237/56\sqrt{-7}-87/8 & 0 & 285/56\sqrt{-7}-87/8 & -87/56\sqrt{-7}+37/8 & 3/14\sqrt{-7}+1/2 & -1 & -111/56\sqrt{-7}+29/8 & -3/2\sqrt{-7}+5/2 & 0 & -33/14\sqrt{-7}+15/2 & -12/7\sqrt{-7}+5 & -3/7\sqrt{-7}-1 & 0 & 0\\ 0 & 0 & 0 & 0 & 285/56\sqrt{-7}-87/8 & 0 & 0 & 237/56\sqrt{-7}-87/8 & -285/56\sqrt{-7}+87/8 & 0 & -87/56\sqrt{-7}+37/8 & -237/56\sqrt{-7}+87/8 & -111/56\sqrt{-7}+29/8 & 87/56\sqrt{-7}-37/8 & 12/7\sqrt{-7}-5 & 111/56\sqrt{-7}-29/8 & -12/7\sqrt{-7}+5\\ 0 & 0 & 0 & 3/7\sqrt{-7} & 87/56\sqrt{-7}-37/8 & 9/14\sqrt{-7}-5/2 & 0 & 3/7\sqrt{-7}+1 & -9/8\sqrt{-7}+45/8 & -3/7\sqrt{-7} & 285/56\sqrt{-7}-87/8 & 0 & -6/7\sqrt{-7} & -333/56\sqrt{-7}+87/8 & 3/14\sqrt{-7}+1/2 & 0 & 0\\ 0 & 0 & 0 & -285/56\sqrt{-7}+87/8 & 3/14\sqrt{-7}+1/2 & -237/56\sqrt{-7}+87/8 & 111/56\sqrt{-7}-29/8 & 0 & 0 & 87/56\sqrt{-7}-37/8 & 39/14\sqrt{-7}-15/2 & -1 & 3/14\sqrt{-7}-5/2 & 0 & -9/8\sqrt{-7}+45/8 & 12/7\sqrt{-7}-5 & 111/56\sqrt{-7}-29/8\\ 0 & 0 & 0 & 237/56\sqrt{-7}-87/8 & -1 & 285/56\sqrt{-7}-87/8 & -87/56\sqrt{-7}+37/8 & 0 & 0 & -111/56\sqrt{-7}+29/8 & 0 & 3/14\sqrt{-7}+1/2 & -12/7\sqrt{-7}+5 & -45/14\sqrt{-7}+15/2 & 0 & -9/14\sqrt{-7}+5/2 & -3/7\sqrt{-7}-1\\ 0 & 0 & 0 & 9/7\sqrt{-7} & -135/56\sqrt{-7}+21/8 & -3/2\sqrt{-7}+5/2 & -3/7\sqrt{-7} & 0 & 111/56\sqrt{-7}-29/8 & 0 & 0 & -3/7\sqrt{-7}-1 & 333/56\sqrt{-7}-87/8 & 6/7\sqrt{-7} & 0 & -285/56\sqrt{-7}+87/8 & -3/14\sqrt{-7}-1/2\\ 0 & 0 & 0 & 111/56\sqrt{-7}-29/8 & 3/2\sqrt{-7}-5/2 & -99/28\sqrt{-7}+33/4 & -285/56\sqrt{-7}+87/8 & 39/14\sqrt{-7}-15/2 & 12/7\sqrt{-7}-5 & 237/56\sqrt{-7}-87/8 & 0 & 0 & -3/14\sqrt{-7}+1/2 & -1 & -27/8\sqrt{-7}+87/8 & 0 & 0\\ 0 & 0 & 0 & -285/56\sqrt{-7}+87/8 & 0 & -237/56\sqrt{-7}+87/8 & 111/56\sqrt{-7}-29/8 & -1 & 3/14\sqrt{-7}+1/2 & 87/56\sqrt{-7}-37/8 & 12/7\sqrt{-7}-5 & 0 & 0 & 15/14\sqrt{-7}-5/2 & -87/56\sqrt{-7}+37/8 & 27/14\sqrt{-7}-15/2 & 135/56\sqrt{-7}-21/8\\ 0 & 0 & 0 & -87/56\sqrt{-7}+37/8 & -33/14\sqrt{-7}+15/2 & 99/28\sqrt{-7}-33/4 & 237/56\sqrt{-7}-87/8 & -3/14\sqrt{-7}+5/2 & 0 & -285/56\sqrt{-7}+87/8 & -3/14\sqrt{-7}+1/2 & -12/7\sqrt{-7}+5 & 0 & 0 & -6/7\sqrt{-7} & -1 & 237/56\sqrt{-7}-87/8\\ 0 & 0 & 0 & 111/56\sqrt{-7}-29/8 & 0 & -99/28\sqrt{-7}+33/4 & -285/56\sqrt{-7}+87/8 & 12/7\sqrt{-7}-5 & 45/14\sqrt{-7}-15/2 & 237/56\sqrt{-7}-87/8 & -1 & 15/14\sqrt{-7}-5/2 & 0 & 0 & -237/56\sqrt{-7}+87/8 & -3/14\sqrt{-7}+1/2 & 6/7\sqrt{-7}\\ 0 & 0 & 0 & 0 & -135/56\sqrt{-7}+21/8 & -12/7\sqrt{-7}+5 & 0 & 9/8\sqrt{-7}-45/8 & 87/56\sqrt{-7}-37/8 & 0 & -27/8\sqrt{-7}+87/8 & -111/56\sqrt{-7}+29/8 & 333/56\sqrt{-7}-87/8 & 285/56\sqrt{-7}-87/8 & 0 & -237/56\sqrt{-7}+87/8 & -1\\ 0 & 0 & 0 & -87/56\sqrt{-7}+37/8 & -12/7\sqrt{-7}+5 & 99/28\sqrt{-7}-33/4 & 237/56\sqrt{-7}-87/8 & 0 & -9/14\sqrt{-7}+5/2 & -285/56\sqrt{-7}+87/8 & 0 & -27/14\sqrt{-7}+15/2 & -1 & -3/14\sqrt{-7}+1/2 & 0 & 0 & 27/8\sqrt{-7}-87/8\\ 0 & 0 & 0 & 0 & -111/56\sqrt{-7}+29/8 & -12/7\sqrt{-7}+5 & 0 & 87/56\sqrt{-7}-37/8 & 9/8\sqrt{-7}-45/8 & 0 & -237/56\sqrt{-7}+87/8 & -135/56\sqrt{-7}+21/8 & 285/56\sqrt{-7}-87/8 & 333/56\sqrt{-7}-87/8 & -1 & -27/8\sqrt{-7}+87/8 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
877g4237287g36+877g34+29287g2937287g28+277g26+79287g22+29287g21+377g20+277g1895287g15+79287g147g13+177g12477g1095287g7177g6577g5+177g4477g2+477g2+177g4577g5177g6+95287g7477g10177g12+7g13+79287g1495287g15277g18+377g2029287g2179287g22+277g2637287g28+29287g29+877g34+37287g36877g428/7\sqrt{-7}g_{42}-37/28\sqrt{-7}g_{36}+8/7\sqrt{-7}g_{34}+29/28\sqrt{-7}g_{29}-37/28\sqrt{-7}g_{28}+2/7\sqrt{-7}g_{26}+79/28\sqrt{-7}g_{22}+29/28\sqrt{-7}g_{21}+3/7\sqrt{-7}g_{20}+2/7\sqrt{-7}g_{18}-95/28\sqrt{-7}g_{15}+79/28\sqrt{-7}g_{14}-\sqrt{-7}g_{13}+1/7\sqrt{-7}g_{12}-4/7\sqrt{-7}g_{10}-95/28\sqrt{-7}g_{7}-1/7\sqrt{-7}g_{6}-5/7\sqrt{-7}g_{5}+1/7\sqrt{-7}g_{4}-4/7\sqrt{-7}g_{2}+4/7\sqrt{-7}g_{-2}+1/7\sqrt{-7}g_{-4}-5/7\sqrt{-7}g_{-5}-1/7\sqrt{-7}g_{-6}+95/28\sqrt{-7}g_{-7}-4/7\sqrt{-7}g_{-10}-1/7\sqrt{-7}g_{-12}+\sqrt{-7}g_{-13}+79/28\sqrt{-7}g_{-14}-95/28\sqrt{-7}g_{-15}-2/7\sqrt{-7}g_{-18}+3/7\sqrt{-7}g_{-20}-29/28\sqrt{-7}g_{-21}-79/28\sqrt{-7}g_{-22}+2/7\sqrt{-7}g_{-26}-37/28\sqrt{-7}g_{-28}+29/28\sqrt{-7}g_{-29}+8/7\sqrt{-7}g_{-34}+37/28\sqrt{-7}g_{-36}-8/7\sqrt{-7}g_{-42}
matching e: g42+(153273732)g36+g34+(7327+2932)g29+(153273732)g28+(147+14)g26+(11167+116)g22+(7327+2932)g21+(1167+1316)g20+(147+14)g18+(11167916)g15+(11167+116)g14+(1167716)g13+(1167+916)g1212g10+(11167916)g7+(18718)g6+(1167316)g5+(187+18)g412g2+12g2+(187+18)g4+(1167316)g5+(18718)g6+(11167+916)g712g10+(1167916)g12+(1167+716)g13+(11167+116)g14+(11167916)g15+(14714)g18+(1167+1316)g20+(73272932)g21+(11167116)g22+(147+14)g26+(153273732)g28+(7327+2932)g29+g34+(15327+3732)g36g42g_{42}+\left(15/32\sqrt{-7}-37/32\right)g_{36}+g_{34}+\left(-7/32\sqrt{-7}+29/32\right)g_{29}+\left(15/32\sqrt{-7}-37/32\right)g_{28}+\left(-1/4\sqrt{-7}+1/4\right)g_{26}+\left(-11/16\sqrt{-7}+1/16\right)g_{22}+\left(-7/32\sqrt{-7}+29/32\right)g_{21}+\left(1/16\sqrt{-7}+13/16\right)g_{20}+\left(-1/4\sqrt{-7}+1/4\right)g_{18}+\left(11/16\sqrt{-7}-9/16\right)g_{15}+\left(-11/16\sqrt{-7}+1/16\right)g_{14}+\left(1/16\sqrt{-7}-7/16\right)g_{13}+\left(1/16\sqrt{-7}+9/16\right)g_{12}-1/2g_{10}+\left(11/16\sqrt{-7}-9/16\right)g_{7}+\left(1/8\sqrt{-7}-1/8\right)g_{6}+\left(1/16\sqrt{-7}-3/16\right)g_{5}+\left(-1/8\sqrt{-7}+1/8\right)g_{4}-1/2g_{2}+1/2g_{-2}+\left(-1/8\sqrt{-7}+1/8\right)g_{-4}+\left(1/16\sqrt{-7}-3/16\right)g_{-5}+\left(1/8\sqrt{-7}-1/8\right)g_{-6}+\left(-11/16\sqrt{-7}+9/16\right)g_{-7}-1/2g_{-10}+\left(-1/16\sqrt{-7}-9/16\right)g_{-12}+\left(-1/16\sqrt{-7}+7/16\right)g_{-13}+\left(-11/16\sqrt{-7}+1/16\right)g_{-14}+\left(11/16\sqrt{-7}-9/16\right)g_{-15}+\left(1/4\sqrt{-7}-1/4\right)g_{-18}+\left(1/16\sqrt{-7}+13/16\right)g_{-20}+\left(7/32\sqrt{-7}-29/32\right)g_{-21}+\left(11/16\sqrt{-7}-1/16\right)g_{-22}+\left(-1/4\sqrt{-7}+1/4\right)g_{-26}+\left(15/32\sqrt{-7}-37/32\right)g_{-28}+\left(-7/32\sqrt{-7}+29/32\right)g_{-29}+g_{-34}+\left(-15/32\sqrt{-7}+37/32\right)g_{-36}-g_{-42}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action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begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -79/28\sqrt{-7} & 0 & -2/7\sqrt{-7} & -95/28\sqrt{-7} & 79/28\sqrt{-7} & 6/7\sqrt{-7} & -2/7\sqrt{-7} & 95/28\sqrt{-7} & 2/7\sqrt{-7} & 2/7\sqrt{-7} & -\sqrt{-7} & -2/7\sqrt{-7} & 3/7\sqrt{-7}\\ 0 & 0 & 0 & -79/28\sqrt{-7} & 0 & -95/28\sqrt{-7} & 29/28\sqrt{-7} & -1/7\sqrt{-7} & 0 & 37/28\sqrt{-7} & \sqrt{-7} & 0 & 11/7\sqrt{-7} & 8/7\sqrt{-7} & 2/7\sqrt{-7} & 0 & 0\\ 0 & 0 & 0 & 0 & -95/28\sqrt{-7} & 0 & 0 & -79/28\sqrt{-7} & 95/28\sqrt{-7} & 0 & 29/28\sqrt{-7} & 79/28\sqrt{-7} & 37/28\sqrt{-7} & -29/28\sqrt{-7} & -8/7\sqrt{-7} & -37/28\sqrt{-7} & 8/7\sqrt{-7}\\ 0 & 0 & 0 & -2/7\sqrt{-7} & -29/28\sqrt{-7} & -3/7\sqrt{-7} & 0 & -2/7\sqrt{-7} & 3/4\sqrt{-7} & 2/7\sqrt{-7} & -95/28\sqrt{-7} & 0 & 4/7\sqrt{-7} & 111/28\sqrt{-7} & -1/7\sqrt{-7} & 0 & 0\\ 0 & 0 & 0 & 95/28\sqrt{-7} & -1/7\sqrt{-7} & 79/28\sqrt{-7} & -37/28\sqrt{-7} & 0 & 0 & -29/28\sqrt{-7} & -13/7\sqrt{-7} & 0 & -1/7\sqrt{-7} & 0 & 3/4\sqrt{-7} & -8/7\sqrt{-7} & -37/28\sqrt{-7}\\ 0 & 0 & 0 & -79/28\sqrt{-7} & 0 & -95/28\sqrt{-7} & 29/28\sqrt{-7} & 0 & 0 & 37/28\sqrt{-7} & 0 & -1/7\sqrt{-7} & 8/7\sqrt{-7} & 15/7\sqrt{-7} & 0 & 3/7\sqrt{-7} & 2/7\sqrt{-7}\\ 0 & 0 & 0 & -6/7\sqrt{-7} & 45/28\sqrt{-7} & \sqrt{-7} & 2/7\sqrt{-7} & 0 & -37/28\sqrt{-7} & 0 & 0 & 2/7\sqrt{-7} & -111/28\sqrt{-7} & -4/7\sqrt{-7} & 0 & 95/28\sqrt{-7} & 1/7\sqrt{-7}\\ 0 & 0 & 0 & -37/28\sqrt{-7} & -\sqrt{-7} & 33/14\sqrt{-7} & 95/28\sqrt{-7} & -13/7\sqrt{-7} & -8/7\sqrt{-7} & -79/28\sqrt{-7} & 0 & 0 & 1/7\sqrt{-7} & 0 & 9/4\sqrt{-7} & 0 & 0\\ 0 & 0 & 0 & 95/28\sqrt{-7} & 0 & 79/28\sqrt{-7} & -37/28\sqrt{-7} & 0 & -1/7\sqrt{-7} & -29/28\sqrt{-7} & -8/7\sqrt{-7} & 0 & 0 & -5/7\sqrt{-7} & 29/28\sqrt{-7} & -9/7\sqrt{-7} & -45/28\sqrt{-7}\\ 0 & 0 & 0 & 29/28\sqrt{-7} & 11/7\sqrt{-7} & -33/14\sqrt{-7} & -79/28\sqrt{-7} & 1/7\sqrt{-7} & 0 & 95/28\sqrt{-7} & 1/7\sqrt{-7} & 8/7\sqrt{-7} & 0 & 0 & 4/7\sqrt{-7} & 0 & -79/28\sqrt{-7}\\ 0 & 0 & 0 & -37/28\sqrt{-7} & 0 & 33/14\sqrt{-7} & 95/28\sqrt{-7} & -8/7\sqrt{-7} & -15/7\sqrt{-7} & -79/28\sqrt{-7} & 0 & -5/7\sqrt{-7} & 0 & 0 & 79/28\sqrt{-7} & 1/7\sqrt{-7} & -4/7\sqrt{-7}\\ 0 & 0 & 0 & 0 & 45/28\sqrt{-7} & 8/7\sqrt{-7} & 0 & -3/4\sqrt{-7} & -29/28\sqrt{-7} & 0 & 9/4\sqrt{-7} & 37/28\sqrt{-7} & -111/28\sqrt{-7} & -95/28\sqrt{-7} & 0 & 79/28\sqrt{-7} & 0\\ 0 & 0 & 0 & 29/28\sqrt{-7} & 8/7\sqrt{-7} & -33/14\sqrt{-7} & -79/28\sqrt{-7} & 0 & 3/7\sqrt{-7} & 95/28\sqrt{-7} & 0 & 9/7\sqrt{-7} & 0 & 1/7\sqrt{-7} & 0 & 0 & -9/4\sqrt{-7}\\ 0 & 0 & 0 & 0 & 37/28\sqrt{-7} & 8/7\sqrt{-7} & 0 & -29/28\sqrt{-7} & -3/4\sqrt{-7} & 0 & 79/28\sqrt{-7} & 45/28\sqrt{-7} & -95/28\sqrt{-7} & -111/28\sqrt{-7} & 0 & 9/4\sqrt{-7} & 0\\ \end{pmatrix}
(1277+5)g42+(111567298)g36+(1277+5)g34+(87567+378)g29+(111567298)g28+(3771)g26+(237567+878)g22+(87567+378)g21+(9147+52)g20+(3771)g18+(285567878)g15+(237567+878)g14+(32752)g13+(3147+52)g12+677g10+g8+(285567878)g7+(3147+12)g6+(1514752)g5+(3147+12)g4+677g2677g2+(3147+12)g4+(1514752)g5+(3147+12)g6+(285567+878)g7+g8+677g10+(314752)g12+(327+52)g13+(237567+878)g14+(285567878)g15+(377+1)g18+(9147+52)g20+(87567378)g21+(237567878)g22+(3771)g26+(111567298)g28+(87567+378)g29+(1277+5)g34+(111567+298)g36+(12775)g42\left(-12/7\sqrt{-7}+5\right)g_{42}+\left(111/56\sqrt{-7}-29/8\right)g_{36}+\left(-12/7\sqrt{-7}+5\right)g_{34}+\left(-87/56\sqrt{-7}+37/8\right)g_{29}+\left(111/56\sqrt{-7}-29/8\right)g_{28}+\left(-3/7\sqrt{-7}-1\right)g_{26}+\left(-237/56\sqrt{-7}+87/8\right)g_{22}+\left(-87/56\sqrt{-7}+37/8\right)g_{21}+\left(-9/14\sqrt{-7}+5/2\right)g_{20}+\left(-3/7\sqrt{-7}-1\right)g_{18}+\left(285/56\sqrt{-7}-87/8\right)g_{15}+\left(-237/56\sqrt{-7}+87/8\right)g_{14}+\left(3/2\sqrt{-7}-5/2\right)g_{13}+\left(-3/14\sqrt{-7}+5/2\right)g_{12}+6/7\sqrt{-7}g_{10}+g_{8}+\left(285/56\sqrt{-7}-87/8\right)g_{7}+\left(3/14\sqrt{-7}+1/2\right)g_{6}+\left(15/14\sqrt{-7}-5/2\right)g_{5}+\left(-3/14\sqrt{-7}+1/2\right)g_{4}+6/7\sqrt{-7}g_{2}-6/7\sqrt{-7}g_{-2}+\left(-3/14\sqrt{-7}+1/2\right)g_{-4}+\left(15/14\sqrt{-7}-5/2\right)g_{-5}+\left(3/14\sqrt{-7}+1/2\right)g_{-6}+\left(-285/56\sqrt{-7}+87/8\right)g_{-7}+g_{-8}+6/7\sqrt{-7}g_{-10}+\left(3/14\sqrt{-7}-5/2\right)g_{-12}+\left(-3/2\sqrt{-7}+5/2\right)g_{-13}+\left(-237/56\sqrt{-7}+87/8\right)g_{-14}+\left(285/56\sqrt{-7}-87/8\right)g_{-15}+\left(3/7\sqrt{-7}+1\right)g_{-18}+\left(-9/14\sqrt{-7}+5/2\right)g_{-20}+\left(87/56\sqrt{-7}-37/8\right)g_{-21}+\left(237/56\sqrt{-7}-87/8\right)g_{-22}+\left(-3/7\sqrt{-7}-1\right)g_{-26}+\left(111/56\sqrt{-7}-29/8\right)g_{-28}+\left(-87/56\sqrt{-7}+37/8\right)g_{-29}+\left(-12/7\sqrt{-7}+5\right)g_{-34}+\left(-111/56\sqrt{-7}+29/8\right)g_{-36}+\left(12/7\sqrt{-7}-5\right)g_{-42}
matching e: g42+(3652122722852122)g36+g34+(2110617+12071061)g29+(7414244746974244)g28+(657424471314244)g26+(91921227+27012122)g22+(9542447+49554244)g21+(20742447+21974244)g20+(635424473854244)g18+(4871061716681061)g15+(190742447+54274244)g14+(2074244720474244)g13+(20742447+21974244)g12+(414244712454244)g10+(1121227+1272122)g8+(20174244766974244)g7+(1110617+1271061)g6+(2074244720474244)g5+(11212271272122)g4+(1794244712954244)g2+(17942447+12954244)g2+(11212271272122)g4+(2074244720474244)g5+(1110617+1271061)g6+(201742447+66974244)g7+(1121227+1272122)g8+(414244712454244)g10+(2074244721974244)g12+(20742447+20474244)g13+(190742447+54274244)g14+(4871061716681061)g15+(63542447+3854244)g18+(20742447+21974244)g20+(954244749554244)g21+(9192122727012122)g22+(657424471314244)g26+(7414244746974244)g28+(2110617+12071061)g29+g34+(36521227+22852122)g36g42g_{42}+\left(-365/2122\sqrt{-7}-2285/2122\right)g_{36}+g_{34}+\left(21/1061\sqrt{-7}+1207/1061\right)g_{29}+\left(-741/4244\sqrt{-7}-4697/4244\right)g_{28}+\left(657/4244\sqrt{-7}-131/4244\right)g_{26}+\left(919/2122\sqrt{-7}+2701/2122\right)g_{22}+\left(95/4244\sqrt{-7}+4955/4244\right)g_{21}+\left(207/4244\sqrt{-7}+2197/4244\right)g_{20}+\left(635/4244\sqrt{-7}-385/4244\right)g_{18}+\left(-487/1061\sqrt{-7}-1668/1061\right)g_{15}+\left(1907/4244\sqrt{-7}+5427/4244\right)g_{14}+\left(207/4244\sqrt{-7}-2047/4244\right)g_{13}+\left(207/4244\sqrt{-7}+2197/4244\right)g_{12}+\left(-41/4244\sqrt{-7}-1245/4244\right)g_{10}+\left(11/2122\sqrt{-7}+127/2122\right)g_{8}+\left(-2017/4244\sqrt{-7}-6697/4244\right)g_{7}+\left(11/1061\sqrt{-7}+127/1061\right)g_{6}+\left(207/4244\sqrt{-7}-2047/4244\right)g_{5}+\left(-11/2122\sqrt{-7}-127/2122\right)g_{4}+\left(-179/4244\sqrt{-7}-1295/4244\right)g_{2}+\left(179/4244\sqrt{-7}+1295/4244\right)g_{-2}+\left(-11/2122\sqrt{-7}-127/2122\right)g_{-4}+\left(207/4244\sqrt{-7}-2047/4244\right)g_{-5}+\left(11/1061\sqrt{-7}+127/1061\right)g_{-6}+\left(2017/4244\sqrt{-7}+6697/4244\right)g_{-7}+\left(11/2122\sqrt{-7}+127/2122\right)g_{-8}+\left(-41/4244\sqrt{-7}-1245/4244\right)g_{-10}+\left(-207/4244\sqrt{-7}-2197/4244\right)g_{-12}+\left(-207/4244\sqrt{-7}+2047/4244\right)g_{-13}+\left(1907/4244\sqrt{-7}+5427/4244\right)g_{-14}+\left(-487/1061\sqrt{-7}-1668/1061\right)g_{-15}+\left(-635/4244\sqrt{-7}+385/4244\right)g_{-18}+\left(207/4244\sqrt{-7}+2197/4244\right)g_{-20}+\left(-95/4244\sqrt{-7}-4955/4244\right)g_{-21}+\left(-919/2122\sqrt{-7}-2701/2122\right)g_{-22}+\left(657/4244\sqrt{-7}-131/4244\right)g_{-26}+\left(-741/4244\sqrt{-7}-4697/4244\right)g_{-28}+\left(21/1061\sqrt{-7}+1207/1061\right)g_{-29}+g_{-34}+\left(365/2122\sqrt{-7}+2285/2122\right)g_{-36}-g_{-42}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (00000000000000000000000000000000000000000000000000000002375678780377285567878237567+878977377+1285567+8783771377132752377+19147+52000237567878028556787887567+3783147+121111567+298327+52033147+1521277+5377100000028556787800237567878285567+878087567+378237567+878111567+29887567378127751115672981277+5000377875673789147520377+1987+4583772855678780677333567+8783147+1200000285567+8783147+12237567+87811156729800875673783914715213147520987+45812775111567298000237567878128556787887567+37800111567+29803147+121277+545147+15209147+523771000977135567+218327+5237701115672980037713335678786770285567+8783147120001115672983275299287+334285567+8783914715212775237567878003147+1212787+87800000285567+8780237567+87811156729813147+12875673781277500151475287567+3782714715213556721800087567+37833147+152992873342375678783147+520285567+8783147+121277+5006771237567878000111567298099287+334285567+87812775451471522375678781151475200237567+8783147+126770000135567+2181277+509874588756737802787+878111567+2983335678782855678780237567+878100087567+3781277+59928733423756787809147+52285567+878027147+15213147+120027878780000111567+2981277+50875673789874580237567+878135567+21828556787833356787812787+8780)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 237/56\sqrt{-7}-87/8 & 0 & 3/7\sqrt{-7} & 285/56\sqrt{-7}-87/8 & -237/56\sqrt{-7}+87/8 & -9/7\sqrt{-7} & 3/7\sqrt{-7}+1 & -285/56\sqrt{-7}+87/8 & -3/7\sqrt{-7}-1 & -3/7\sqrt{-7}-1 & 3/2\sqrt{-7}-5/2 & 3/7\sqrt{-7}+1 & -9/14\sqrt{-7}+5/2\\ 0 & 0 & 0 & 237/56\sqrt{-7}-87/8 & 0 & 285/56\sqrt{-7}-87/8 & -87/56\sqrt{-7}+37/8 & 3/14\sqrt{-7}+1/2 & -1 & -111/56\sqrt{-7}+29/8 & -3/2\sqrt{-7}+5/2 & 0 & -33/14\sqrt{-7}+15/2 & -12/7\sqrt{-7}+5 & -3/7\sqrt{-7}-1 & 0 & 0\\ 0 & 0 & 0 & 0 & 285/56\sqrt{-7}-87/8 & 0 & 0 & 237/56\sqrt{-7}-87/8 & -285/56\sqrt{-7}+87/8 & 0 & -87/56\sqrt{-7}+37/8 & -237/56\sqrt{-7}+87/8 & -111/56\sqrt{-7}+29/8 & 87/56\sqrt{-7}-37/8 & 12/7\sqrt{-7}-5 & 111/56\sqrt{-7}-29/8 & -12/7\sqrt{-7}+5\\ 0 & 0 & 0 & 3/7\sqrt{-7} & 87/56\sqrt{-7}-37/8 & 9/14\sqrt{-7}-5/2 & 0 & 3/7\sqrt{-7}+1 & -9/8\sqrt{-7}+45/8 & -3/7\sqrt{-7} & 285/56\sqrt{-7}-87/8 & 0 & -6/7\sqrt{-7} & -333/56\sqrt{-7}+87/8 & 3/14\sqrt{-7}+1/2 & 0 & 0\\ 0 & 0 & 0 & -285/56\sqrt{-7}+87/8 & 3/14\sqrt{-7}+1/2 & -237/56\sqrt{-7}+87/8 & 111/56\sqrt{-7}-29/8 & 0 & 0 & 87/56\sqrt{-7}-37/8 & 39/14\sqrt{-7}-15/2 & -1 & 3/14\sqrt{-7}-5/2 & 0 & -9/8\sqrt{-7}+45/8 & 12/7\sqrt{-7}-5 & 111/56\sqrt{-7}-29/8\\ 0 & 0 & 0 & 237/56\sqrt{-7}-87/8 & -1 & 285/56\sqrt{-7}-87/8 & -87/56\sqrt{-7}+37/8 & 0 & 0 & -111/56\sqrt{-7}+29/8 & 0 & 3/14\sqrt{-7}+1/2 & -12/7\sqrt{-7}+5 & -45/14\sqrt{-7}+15/2 & 0 & -9/14\sqrt{-7}+5/2 & -3/7\sqrt{-7}-1\\ 0 & 0 & 0 & 9/7\sqrt{-7} & -135/56\sqrt{-7}+21/8 & -3/2\sqrt{-7}+5/2 & -3/7\sqrt{-7} & 0 & 111/56\sqrt{-7}-29/8 & 0 & 0 & -3/7\sqrt{-7}-1 & 333/56\sqrt{-7}-87/8 & 6/7\sqrt{-7} & 0 & -285/56\sqrt{-7}+87/8 & -3/14\sqrt{-7}-1/2\\ 0 & 0 & 0 & 111/56\sqrt{-7}-29/8 & 3/2\sqrt{-7}-5/2 & -99/28\sqrt{-7}+33/4 & -285/56\sqrt{-7}+87/8 & 39/14\sqrt{-7}-15/2 & 12/7\sqrt{-7}-5 & 237/56\sqrt{-7}-87/8 & 0 & 0 & -3/14\sqrt{-7}+1/2 & -1 & -27/8\sqrt{-7}+87/8 & 0 & 0\\ 0 & 0 & 0 & -285/56\sqrt{-7}+87/8 & 0 & -237/56\sqrt{-7}+87/8 & 111/56\sqrt{-7}-29/8 & -1 & 3/14\sqrt{-7}+1/2 & 87/56\sqrt{-7}-37/8 & 12/7\sqrt{-7}-5 & 0 & 0 & 15/14\sqrt{-7}-5/2 & -87/56\sqrt{-7}+37/8 & 27/14\sqrt{-7}-15/2 & 135/56\sqrt{-7}-21/8\\ 0 & 0 & 0 & -87/56\sqrt{-7}+37/8 & -33/14\sqrt{-7}+15/2 & 99/28\sqrt{-7}-33/4 & 237/56\sqrt{-7}-87/8 & -3/14\sqrt{-7}+5/2 & 0 & -285/56\sqrt{-7}+87/8 & -3/14\sqrt{-7}+1/2 & -12/7\sqrt{-7}+5 & 0 & 0 & -6/7\sqrt{-7} & -1 & 237/56\sqrt{-7}-87/8\\ 0 & 0 & 0 & 111/56\sqrt{-7}-29/8 & 0 & -99/28\sqrt{-7}+33/4 & -285/56\sqrt{-7}+87/8 & 12/7\sqrt{-7}-5 & 45/14\sqrt{-7}-15/2 & 237/56\sqrt{-7}-87/8 & -1 & 15/14\sqrt{-7}-5/2 & 0 & 0 & -237/56\sqrt{-7}+87/8 & -3/14\sqrt{-7}+1/2 & 6/7\sqrt{-7}\\ 0 & 0 & 0 & 0 & -135/56\sqrt{-7}+21/8 & -12/7\sqrt{-7}+5 & 0 & 9/8\sqrt{-7}-45/8 & 87/56\sqrt{-7}-37/8 & 0 & -27/8\sqrt{-7}+87/8 & -111/56\sqrt{-7}+29/8 & 333/56\sqrt{-7}-87/8 & 285/56\sqrt{-7}-87/8 & 0 & -237/56\sqrt{-7}+87/8 & -1\\ 0 & 0 & 0 & -87/56\sqrt{-7}+37/8 & -12/7\sqrt{-7}+5 & 99/28\sqrt{-7}-33/4 & 237/56\sqrt{-7}-87/8 & 0 & -9/14\sqrt{-7}+5/2 & -285/56\sqrt{-7}+87/8 & 0 & -27/14\sqrt{-7}+15/2 & -1 & -3/14\sqrt{-7}+1/2 & 0 & 0 & 27/8\sqrt{-7}-87/8\\ 0 & 0 & 0 & 0 & -111/56\sqrt{-7}+29/8 & -12/7\sqrt{-7}+5 & 0 & 87/56\sqrt{-7}-37/8 & 9/8\sqrt{-7}-45/8 & 0 & -237/56\sqrt{-7}+87/8 & -135/56\sqrt{-7}+21/8 & 285/56\sqrt{-7}-87/8 & 333/56\sqrt{-7}-87/8 & -1 & -27/8\sqrt{-7}+87/8 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (3, 2), (-3, -2), (1, 0), (-1, 0), (2, 1), (-2, -1), (1, 1), (-1, -1), (3, 1), (-3, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240360360720)\begin{pmatrix}240 & -360\\ -360 & 720\\ \end{pmatrix}

Scalar product computed: (130)\begin{pmatrix}1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
122g1+122h1+122g11/2\sqrt{2}g_{1}+1/2\sqrt{2}h_{1}+1/2\sqrt{2}g_{-1}
matching e: g1+(2+1)h1+(223)g1g_{1}+\left(-\sqrt{2}+1\right)h_{1}+\left(2\sqrt{2}-3\right)g_{-1}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (22000000000000000122012200000000000000022000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & -1/2\sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
122g1+122h1+122g11/2\sqrt{2}g_{1}+1/2\sqrt{2}h_{1}+1/2\sqrt{2}g_{-1}
matching e: g1+(2+1)h1+(223)g1g_{1}+\left(-\sqrt{2}+1\right)h_{1}+\left(2\sqrt{2}-3\right)g_{-1}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (22000000000000000122012200000000000000022000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & -1/2\sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120)\begin{pmatrix}120\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=4h8+8h7+12h6+16h5+20h4+14h3+10h2+7h1e=x1g106+x2g91+x3g83+x4g72+x6g71+x7g70+x5g69e=x12g69+x14g70+x13g71+x11g72+x10g83+x9g91+x8g106\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+7h_{1}\\ e&=&x_{1} g_{106}+x_{2} g_{91}+x_{3} g_{83}+x_{4} g_{72}+x_{6} g_{71}+x_{7} g_{70}+x_{5} g_{69}\\ f&=&x_{12} g_{-69}+x_{14} g_{-70}+x_{13} g_{-71}+x_{11} g_{-72}+x_{10} g_{-83}+x_{9} g_{-91}+x_{8} g_{-106}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x8+x2x9+x3x10+x4x114)h8+(2x1x8+2x2x9+x3x10+x4x11+x6x13+x7x148)h7+(3x1x8+2x2x9+2x3x10+x4x11+x5x12+2x6x13+x7x1412)h6+(4x1x8+2x2x9+2x3x10+2x4x11+2x5x12+2x6x13+2x7x1416)h5+(4x1x8+3x2x9+3x3x10+2x4x11+3x5x12+2x6x13+3x7x1420)h4+(2x1x8+2x2x9+2x3x10+2x4x11+2x5x12+2x6x13+2x7x1414)h3+(2x1x8+2x2x9+x3x10+x4x11+2x5x12+x6x13+x7x1410)h2+(x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+x7x147)h1[e,f] - h = \left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} -4\right)h_{8}+\left(2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -8\right)h_{7}+\left(3x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -12\right)h_{6}+\left(4x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -16\right)h_{5}+\left(4x_{1} x_{8} +3x_{2} x_{9} +3x_{3} x_{10} +2x_{4} x_{11} +3x_{5} x_{12} +2x_{6} x_{13} +3x_{7} x_{14} -20\right)h_{4}+\left(2x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -14\right)h_{3}+\left(2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -10\right)h_{2}+\left(x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -7\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+x7x147=02x1x8+2x2x9+x3x10+x4x11+2x5x12+x6x13+x7x1410=02x1x8+2x2x9+2x3x10+2x4x11+2x5x12+2x6x13+2x7x1414=04x1x8+3x2x9+3x3x10+2x4x11+3x5x12+2x6x13+3x7x1420=04x1x8+2x2x9+2x3x10+2x4x11+2x5x12+2x6x13+2x7x1416=03x1x8+2x2x9+2x3x10+x4x11+x5x12+2x6x13+x7x1412=02x1x8+2x2x9+x3x10+x4x11+x6x13+x7x148=0x1x8+x2x9+x3x10+x4x114=0\begin{array}{rcl}x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -7&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -10&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -14&=&0\\4x_{1} x_{8} +3x_{2} x_{9} +3x_{3} x_{10} +2x_{4} x_{11} +3x_{5} x_{12} +2x_{6} x_{13} +3x_{7} x_{14} -20&=&0\\4x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -16&=&0\\3x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -12&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -8&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} -4&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=4h8+8h7+12h6+16h5+20h4+14h3+10h2+7h1e=x1g106+x2g91+x3g83+x4g72+x6g71+x7g70+x5g69f=g69+g70+g71+g72+g83+g91+g106\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+7h_{1}\\e&=&x_{1} g_{106}+x_{2} g_{91}+x_{3} g_{83}+x_{4} g_{72}+x_{6} g_{71}+x_{7} g_{70}+x_{5} g_{69}\\f&=&g_{-69}+g_{-70}+g_{-71}+g_{-72}+g_{-83}+g_{-91}+g_{-106}\end{array}
Matrix form of the system we are trying to solve: (11111112211211222222243323234222222322112122110111111000)[col. vect.]=(7101420161284)\begin{pmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1\\ 2 & 2 & 1 & 1 & 2 & 1 & 1\\ 2 & 2 & 2 & 2 & 2 & 2 & 2\\ 4 & 3 & 3 & 2 & 3 & 2 & 3\\ 4 & 2 & 2 & 2 & 2 & 2 & 2\\ 3 & 2 & 2 & 1 & 1 & 2 & 1\\ 2 & 2 & 1 & 1 & 0 & 1 & 1\\ 1 & 1 & 1 & 1 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}7\\ 10\\ 14\\ 20\\ 16\\ 12\\ 8\\ 4\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=4h8+8h7+12h6+16h5+20h4+14h3+10h2+7h1e=x1g106+x2g91+x3g83+x4g72+x6g71+x7g70+x5g69f=x12g69+x14g70+x13g71+x11g72+x10g83+x9g91+x8g106\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+16h_{5}+20h_{4}+14h_{3}+10h_{2}+7h_{1}\\ e&=&x_{1} g_{106}+x_{2} g_{91}+x_{3} g_{83}+x_{4} g_{72}+x_{6} g_{71}+x_{7} g_{70}+x_{5} g_{69}\\ f&=&x_{12} g_{-69}+x_{14} g_{-70}+x_{13} g_{-71}+x_{11} g_{-72}+x_{10} g_{-83}+x_{9} g_{-91}+x_{8} g_{-106}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x8+x2x9+x3x10+x4x11+x5x12+x6x13+x7x147=02x1x8+2x2x9+x3x10+x4x11+2x5x12+x6x13+x7x1410=02x1x8+2x2x9+2x3x10+2x4x11+2x5x12+2x6x13+2x7x1414=04x1x8+3x2x9+3x3x10+2x4x11+3x5x12+2x6x13+3x7x1420=04x1x8+2x2x9+2x3x10+2x4x11+2x5x12+2x6x13+2x7x1416=03x1x8+2x2x9+2x3x10+x4x11+x5x12+2x6x13+x7x1412=02x1x8+2x2x9+x3x10+x4x11+x6x13+x7x148=0x1x8+x2x9+x3x10+x4x114=0\begin{array}{rcl}x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -7&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +2x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -10&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -14&=&0\\4x_{1} x_{8} +3x_{2} x_{9} +3x_{3} x_{10} +2x_{4} x_{11} +3x_{5} x_{12} +2x_{6} x_{13} +3x_{7} x_{14} -20&=&0\\4x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -16&=&0\\3x_{1} x_{8} +2x_{2} x_{9} +2x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +2x_{6} x_{13} +x_{7} x_{14} -12&=&0\\2x_{1} x_{8} +2x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -8&=&0\\x_{1} x_{8} +x_{2} x_{9} +x_{3} x_{10} +x_{4} x_{11} -4&=&0\\\end{array}

A16A^{6}_1
h-characteristic: (0, 0, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 12
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: A2+2A1A^{1}_2+2A^{1}_1 Containing regular semisimple subalgebra number 2: 6A16A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 3V4ψ+16V3ψ+27V2ψ+32Vψ+24V03V_{4\psi}+16V_{3\psi}+27V_{2\psi}+32V_{\psi}+24V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=4h8+8h7+12h6+15h5+18h4+12h3+9h2+6h1e=2g104+g93+2g68+g65f=g65+g68+g93+g104\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+15h_{5}+18h_{4}+12h_{3}+9h_{2}+6h_{1}\\ e&=&2g_{104}+g_{93}+2g_{68}+g_{65}\\ f&=&g_{-65}+g_{-68}+g_{-93}+g_{-104}\end{array}
Lie brackets of the above elements.
h=4h8+8h7+12h6+15h5+18h4+12h3+9h2+6h1e=2g104+g93+2g68+g65f=g65+g68+g93+g104\begin{array}{rcl}[e, f]&=&4h_{8}+8h_{7}+12h_{6}+15h_{5}+18h_{4}+12h_{3}+9h_{2}+6h_{1}\\ [h, e]&=&4g_{104}+2g_{93}+4g_{68}+2g_{65}\\ [h, f]&=&-2g_{-65}-2g_{-68}-2g_{-93}-2g_{-104}\end{array}
Centralizer type: B3+A16B_3+A^{6}_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 24): g18g_{-18}, g12g_{-12}, g10g_{-10}, g5g_{-5}, g4g_{-4}, g2g_{-2}, h2h_{2}, h4h_{4}, h5h_{5}, h7h3h1h_{7}-h_{3}-h_{1}, g2g_{2}, g3g31g_{3}-g_{-31}, g4g_{4}, g5g_{5}, g10g_{10}, g11g25g_{11}-g_{-25}, g12g_{12}, g15+12g1+12g812g44g_{15}+1/2g_{-1}+1/2g_{-8}-1/2g_{-44}, g17+g19g_{17}+g_{-19}, g18g_{18}, g19+g17g_{19}+g_{-17}, g25g11g_{25}-g_{-11}, g31g3g_{31}-g_{-3}, g442g8g1g15g_{44}-2g_{8}-g_{1}-g_{-15}
Basis of centralizer intersected with cartan (dimension: 4): h2-h_{2}, h7h3h1h_{7}-h_{3}-h_{1}, h5-h_{5}, h4-h_{4}
Cartan of centralizer (dimension: 4): h7h3h1h_{7}-h_{3}-h_{1}, h2-h_{2}, h5-h_{5}, h4-h_{4}
Cartan-generating semisimple element: 11h7+9h5+7h411h3h211h111h_{7}+9h_{5}+7h_{4}-11h_{3}-h_{2}-11h_{1}
adjoint action: (19000000000000000000000000280000000000000000000000008000000000000000000000000110000000000000000000000001700000000000000000000000090000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000900000000000000000000000018000000000000000000000000170000000000000000000000001100000000000000000000000080000000000000000000000001000000000000000000000000280000000000000000000000001100000000000000000000000010000000000000000000000000190000000000000000000000001000000000000000000000000010000000000000000000000001800000000000000000000000011)\begin{pmatrix}-19 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -28 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -17 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 17 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 28 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 19 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 18 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -11\\ \end{pmatrix}
Characteristic polynomial ad H: x242246x22+2030551x20981430796x18+281158204591x1649765026290486x14+5479459310138761x12364768125227925656x10+13452010550307204496x8214234249522858050816x6+201141577123316121600x4x^{24}-2246x^{22}+2030551x^{20}-981430796x^{18}+281158204591x^{16}-49765026290486x^{14}+5479459310138761x^{12}-364768125227925656x^{10}+13452010550307204496x^8-214234249522858050816x^6+201141577123316121600x^4
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -28)(x -19)(x -18)(x -17)(x -11)(x -11)(x -10)(x -9)(x -8)(x -1)(x +1)(x +8)(x +9)(x +10)(x +11)(x +11)(x +17)(x +18)(x +19)(x +28)
Eigenvalues of ad H: 00, 2828, 1919, 1818, 1717, 1111, 1010, 99, 88, 11, 1-1, 8-8, 9-9, 10-10, 11-11, 17-17, 18-18, 19-19, 28-28
24 eigenvectors of ad H: 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1), 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
Centralizer type: B^{1}_3+A^{6}_1
Reductive components (2 total):
Scalar product computed: (13016016016016001600130)\begin{pmatrix}1/30 & -1/60 & -1/60\\ -1/60 & 1/60 & 0\\ -1/60 & 0 & 1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (3 total):
h2-h_{2}
matching e: g2g_{-2}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (100000000000000000000000010000000000000000000000001000000000000000000000000000000000000000000000000010000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000001000000000000000000000000100000000000000000000000010000000000000000000000000000000000000000000000000100000000000000000000000010000000000000000000000001000000000000000000000000100000000000000000000000000000000000000000000000000)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h5+h2h_{5}+h_{2}
matching e: g25g11g_{25}-g_{-11}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000000020000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000200000000000000000000000020000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h4+h2h_{4}+h_{2}
matching e: g10g_{10}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (100000000000000000000000000000000000000000000000002000000000000000000000000100000000000000000000000010000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000001000000000000000000000000100000000000000000000000010000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000010000000000000000000000001000000000000000000000000000000000000000000000000010000000000000000000000000)\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
h2-h_{2}
matching e: g2g_{-2}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (100000000000000000000000010000000000000000000000001000000000000000000000000000000000000000000000000010000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000001000000000000000000000000100000000000000000000000010000000000000000000000000000000000000000000000000100000000000000000000000010000000000000000000000001000000000000000000000000100000000000000000000000000000000000000000000000000)\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h5+h2h_{5}+h_{2}
matching e: g25g11g_{25}-g_{-11}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000000020000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000200000000000000000000000020000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
h4+h2h_{4}+h_{2}
matching e: g10g_{10}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (100000000000000000000000000000000000000000000000002000000000000000000000000100000000000000000000000010000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000001000000000000000000000000100000000000000000000000010000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000010000000000000000000000001000000000000000000000000000000000000000000000000010000000000000000000000000)\begin{pmatrix}-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}
Elements in Cartan dual to root system: (2, 1, 1), (-2, -1, -1), (1, 1, 1), (-1, -1, -1), (2, 1, 2), (-2, -1, -2), (1, 0, 1), (-1, 0, -1), (1, 1, 0), (-1, -1, 0), (2, 1, 0), (-2, -1, 0), (1, 0, 0), (-1, 0, 0), (0, 0, 1), (0, 0, -1), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120120601202400600120)\begin{pmatrix}120 & -120 & -60\\ -120 & 240 & 0\\ -60 & 0 & 120\\ \end{pmatrix}

Scalar product computed: (1180)\begin{pmatrix}1/180\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
2h7h52h42h3h22h12h_{7}-h_{5}-2h_{4}-2h_{3}-h_{2}-2h_{1}
matching e: g15+12g1+12g812g44g_{15}+1/2g_{-1}+1/2g_{-8}-1/2g_{-44}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
Linear space basis of intersection of centralizer and ambient Cartan:
2h7h52h42h3h22h12h_{7}-h_{5}-2h_{4}-2h_{3}-h_{2}-2h_{1}
matching e: g15+12g1+12g812g44g_{15}+1/2g_{-1}+1/2g_{-8}-1/2g_{-44}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}
Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (720)\begin{pmatrix}720\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=4h8+8h7+12h6+15h5+18h4+12h3+9h2+6h1e=x1g104+x3g93+x2g68+x4g65e=x8g65+x6g68+x7g93+x5g104\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+15h_{5}+18h_{4}+12h_{3}+9h_{2}+6h_{1}\\ e&=&x_{1} g_{104}+x_{3} g_{93}+x_{2} g_{68}+x_{4} g_{65}\\ f&=&x_{8} g_{-65}+x_{6} g_{-68}+x_{7} g_{-93}+x_{5} g_{-104}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x5+x2x64)h8+(2x1x5+x2x6+x3x7+x4x88)h7+(2x1x5+2x2x6+2x3x7+2x4x812)h6+(3x1x5+2x2x6+3x3x7+2x4x815)h5+(4x1x5+2x2x6+4x3x7+2x4x818)h4+(3x1x5+x2x6+3x3x7+x4x812)h3+(2x1x5+x2x6+2x3x7+x4x89)h2+(2x1x5+x3x7+x4x86)h1[e,f] - h = \left(x_{1} x_{5} +x_{2} x_{6} -4\right)h_{8}+\left(2x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -8\right)h_{7}+\left(2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12\right)h_{6}+\left(3x_{1} x_{5} +2x_{2} x_{6} +3x_{3} x_{7} +2x_{4} x_{8} -15\right)h_{5}+\left(4x_{1} x_{5} +2x_{2} x_{6} +4x_{3} x_{7} +2x_{4} x_{8} -18\right)h_{4}+\left(3x_{1} x_{5} +x_{2} x_{6} +3x_{3} x_{7} +x_{4} x_{8} -12\right)h_{3}+\left(2x_{1} x_{5} +x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -9\right)h_{2}+\left(2x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -6\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
2x1x5+x3x7+x4x86=02x1x5+x2x6+2x3x7+x4x89=03x1x5+x2x6+3x3x7+x4x812=04x1x5+2x2x6+4x3x7+2x4x818=03x1x5+2x2x6+3x3x7+2x4x815=02x1x5+2x2x6+2x3x7+2x4x812=02x1x5+x2x6+x3x7+x4x88=0x1x5+x2x64=0\begin{array}{rcl}2x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -6&=&0\\2x_{1} x_{5} +x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -9&=&0\\3x_{1} x_{5} +x_{2} x_{6} +3x_{3} x_{7} +x_{4} x_{8} -12&=&0\\4x_{1} x_{5} +2x_{2} x_{6} +4x_{3} x_{7} +2x_{4} x_{8} -18&=&0\\3x_{1} x_{5} +2x_{2} x_{6} +3x_{3} x_{7} +2x_{4} x_{8} -15&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -8&=&0\\x_{1} x_{5} +x_{2} x_{6} -4&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=4h8+8h7+12h6+15h5+18h4+12h3+9h2+6h1e=x1g104+x3g93+x2g68+x4g65f=g65+g68+g93+g104\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+15h_{5}+18h_{4}+12h_{3}+9h_{2}+6h_{1}\\e&=&x_{1} g_{104}+x_{3} g_{93}+x_{2} g_{68}+x_{4} g_{65}\\f&=&g_{-65}+g_{-68}+g_{-93}+g_{-104}\end{array}
Matrix form of the system we are trying to solve: (20112121313142423232222221111100)[col. vect.]=(691218151284)\begin{pmatrix}2 & 0 & 1 & 1\\ 2 & 1 & 2 & 1\\ 3 & 1 & 3 & 1\\ 4 & 2 & 4 & 2\\ 3 & 2 & 3 & 2\\ 2 & 2 & 2 & 2\\ 2 & 1 & 1 & 1\\ 1 & 1 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}6\\ 9\\ 12\\ 18\\ 15\\ 12\\ 8\\ 4\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=4h8+8h7+12h6+15h5+18h4+12h3+9h2+6h1e=x1g104+x3g93+x2g68+x4g65f=x8g65+x6g68+x7g93+x5g104\begin{array}{rcl}h&=&4h_{8}+8h_{7}+12h_{6}+15h_{5}+18h_{4}+12h_{3}+9h_{2}+6h_{1}\\ e&=&x_{1} g_{104}+x_{3} g_{93}+x_{2} g_{68}+x_{4} g_{65}\\ f&=&x_{8} g_{-65}+x_{6} g_{-68}+x_{7} g_{-93}+x_{5} g_{-104}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
2x1x5+x3x7+x4x86=02x1x5+x2x6+2x3x7+x4x89=03x1x5+x2x6+3x3x7+x4x812=04x1x5+2x2x6+4x3x7+2x4x818=03x1x5+2x2x6+3x3x7+2x4x815=02x1x5+2x2x6+2x3x7+2x4x812=02x1x5+x2x6+x3x7+x4x88=0x1x5+x2x64=0\begin{array}{rcl}2x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -6&=&0\\2x_{1} x_{5} +x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -9&=&0\\3x_{1} x_{5} +x_{2} x_{6} +3x_{3} x_{7} +x_{4} x_{8} -12&=&0\\4x_{1} x_{5} +2x_{2} x_{6} +4x_{3} x_{7} +2x_{4} x_{8} -18&=&0\\3x_{1} x_{5} +2x_{2} x_{6} +3x_{3} x_{7} +2x_{4} x_{8} -15&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -8&=&0\\x_{1} x_{5} +x_{2} x_{6} -4&=&0\\\end{array}

A15A^{5}_1
h-characteristic: (1, 0, 0, 0, 0, 0, 0, 1)
Length of the weight dual to h: 10
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 5A15A^{1}_1 Containing regular semisimple subalgebra number 2: A2+A1A^{1}_2+A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V4ψ+12V3ψ+32V2ψ+32Vψ+35V0V_{4\psi}+12V_{3\psi}+32V_{2\psi}+32V_{\psi}+35V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=4h8+7h7+10h6+13h5+16h4+11h3+8h2+6h1e=g112+g97+g96+g77+g55f=g55+g77+g96+g97+g112\begin{array}{rcl}h&=&4h_{8}+7h_{7}+10h_{6}+13h_{5}+16h_{4}+11h_{3}+8h_{2}+6h_{1}\\ e&=&g_{112}+g_{97}+g_{96}+g_{77}+g_{55}\\ f&=&g_{-55}+g_{-77}+g_{-96}+g_{-97}+g_{-112}\end{array}
Lie brackets of the above elements.
h=4h8+7h7+10h6+13h5+16h4+11h3+8h2+6h1e=g112+g97+g96+g77+g55f=g55+g77+g96+g97+g112\begin{array}{rcl}[e, f]&=&4h_{8}+7h_{7}+10h_{6}+13h_{5}+16h_{4}+11h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&2g_{112}+2g_{97}+2g_{96}+2g_{77}+2g_{55}\\ [h, f]&=&-2g_{-55}-2g_{-77}-2g_{-96}-2g_{-97}-2g_{-112}\end{array}
Centralizer type: A5A_5
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Unknown elements.
h=4h8+7h7+10h6+13h5+16h4+11h3+8h2+6h1e=x1g112+x5g97+x2g96+x3g77+x4g55e=x9g55+x8g77+x7g96+x10g97+x6g112\begin{array}{rcl}h&=&4h_{8}+7h_{7}+10h_{6}+13h_{5}+16h_{4}+11h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{112}+x_{5} g_{97}+x_{2} g_{96}+x_{3} g_{77}+x_{4} g_{55}\\ f&=&x_{9} g_{-55}+x_{8} g_{-77}+x_{7} g_{-96}+x_{10} g_{-97}+x_{6} g_{-112}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x6+x2x7+x3x8+x4x94)h8+(2x1x6+2x2x7+x3x8+x4x9+x5x107)h7+(3x1x6+3x2x7+x3x8+x4x9+2x5x1010)h6+(4x1x6+3x2x7+2x3x8+x4x9+3x5x1013)h5+(5x1x6+3x2x7+3x3x8+x4x9+4x5x1016)h4+(3x1x6+2x2x7+2x3x8+x4x9+3x5x1011)h3+(3x1x6+x2x7+x3x8+x4x9+2x5x108)h2+(x1x6+x2x7+x3x8+x4x9+2x5x106)h1[e,f] - h = \left(x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} -4\right)h_{8}+\left(2x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -7\right)h_{7}+\left(3x_{1} x_{6} +3x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -10\right)h_{6}+\left(4x_{1} x_{6} +3x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +3x_{5} x_{10} -13\right)h_{5}+\left(5x_{1} x_{6} +3x_{2} x_{7} +3x_{3} x_{8} +x_{4} x_{9} +4x_{5} x_{10} -16\right)h_{4}+\left(3x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +3x_{5} x_{10} -11\right)h_{3}+\left(3x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -8\right)h_{2}+\left(x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -6\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
x1x6+x2x7+x3x8+x4x9+2x5x106=03x1x6+x2x7+x3x8+x4x9+2x5x108=03x1x6+2x2x7+2x3x8+x4x9+3x5x1011=05x1x6+3x2x7+3x3x8+x4x9+4x5x1016=04x1x6+3x2x7+2x3x8+x4x9+3x5x1013=03x1x6+3x2x7+x3x8+x4x9+2x5x1010=02x1x6+2x2x7+x3x8+x4x9+x5x107=0x1x6+x2x7+x3x8+x4x94=0\begin{array}{rcl}x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -6&=&0\\3x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -8&=&0\\3x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +3x_{5} x_{10} -11&=&0\\5x_{1} x_{6} +3x_{2} x_{7} +3x_{3} x_{8} +x_{4} x_{9} +4x_{5} x_{10} -16&=&0\\4x_{1} x_{6} +3x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +3x_{5} x_{10} -13&=&0\\3x_{1} x_{6} +3x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -10&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -7&=&0\\x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} -4&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=4h8+7h7+10h6+13h5+16h4+11h3+8h2+6h1e=x1g112+x5g97+x2g96+x3g77+x4g55f=g55+g77+g96+g97+g112\begin{array}{rcl}h&=&4h_{8}+7h_{7}+10h_{6}+13h_{5}+16h_{4}+11h_{3}+8h_{2}+6h_{1}\\e&=&x_{1} g_{112}+x_{5} g_{97}+x_{2} g_{96}+x_{3} g_{77}+x_{4} g_{55}\\f&=&g_{-55}+g_{-77}+g_{-96}+g_{-97}+g_{-112}\end{array}
Matrix form of the system we are trying to solve: (1111231112322135331443213331122211111110)[col. vect.]=(681116131074)\begin{pmatrix}1 & 1 & 1 & 1 & 2\\ 3 & 1 & 1 & 1 & 2\\ 3 & 2 & 2 & 1 & 3\\ 5 & 3 & 3 & 1 & 4\\ 4 & 3 & 2 & 1 & 3\\ 3 & 3 & 1 & 1 & 2\\ 2 & 2 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}6\\ 8\\ 11\\ 16\\ 13\\ 10\\ 7\\ 4\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=4h8+7h7+10h6+13h5+16h4+11h3+8h2+6h1e=x1g112+x5g97+x2g96+x3g77+x4g55f=x9g55+x8g77+x7g96+x10g97+x6g112\begin{array}{rcl}h&=&4h_{8}+7h_{7}+10h_{6}+13h_{5}+16h_{4}+11h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{112}+x_{5} g_{97}+x_{2} g_{96}+x_{3} g_{77}+x_{4} g_{55}\\ f&=&x_{9} g_{-55}+x_{8} g_{-77}+x_{7} g_{-96}+x_{10} g_{-97}+x_{6} g_{-112}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
x1x6+x2x7+x3x8+x4x9+2x5x106=03x1x6+x2x7+x3x8+x4x9+2x5x108=03x1x6+2x2x7+2x3x8+x4x9+3x5x1011=05x1x6+3x2x7+3x3x8+x4x9+4x5x1016=04x1x6+3x2x7+2x3x8+x4x9+3x5x1013=03x1x6+3x2x7+x3x8+x4x9+2x5x1010=02x1x6+2x2x7+x3x8+x4x9+x5x107=0x1x6+x2x7+x3x8+x4x94=0\begin{array}{rcl}x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -6&=&0\\3x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -8&=&0\\3x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +3x_{5} x_{10} -11&=&0\\5x_{1} x_{6} +3x_{2} x_{7} +3x_{3} x_{8} +x_{4} x_{9} +4x_{5} x_{10} -16&=&0\\4x_{1} x_{6} +3x_{2} x_{7} +2x_{3} x_{8} +x_{4} x_{9} +3x_{5} x_{10} -13&=&0\\3x_{1} x_{6} +3x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +2x_{5} x_{10} -10&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -7&=&0\\x_{1} x_{6} +x_{2} x_{7} +x_{3} x_{8} +x_{4} x_{9} -4&=&0\\\end{array}

A14A^{4}_1
h-characteristic: (0, 0, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1: 4A14A^{1}_1 Containing regular semisimple subalgebra number 2: A2A^{1}_2
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V4ψ+55V2ψ+78V0V_{4\psi}+55V_{2\psi}+78V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=4h8+6h7+8h6+10h5+12h4+8h3+6h2+4h1e=g119+g101+g68+g15f=g15+g68+g101+g119\begin{array}{rcl}h&=&4h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&g_{119}+g_{101}+g_{68}+g_{15}\\ f&=&g_{-15}+g_{-68}+g_{-101}+g_{-119}\end{array}
Lie brackets of the above elements.
h=4h8+6h7+8h6+10h5+12h4+8h3+6h2+4h1e=g119+g101+g68+g15f=g15+g68+g101+g119\begin{array}{rcl}[e, f]&=&4h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&2g_{119}+2g_{101}+2g_{68}+2g_{15}\\ [h, f]&=&-2g_{-15}-2g_{-68}-2g_{-101}-2g_{-119}\end{array}
Centralizer type: E6E_6
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Unknown elements.
h=4h8+6h7+8h6+10h5+12h4+8h3+6h2+4h1e=x1g119+x2g101+x3g68+x4g15e=x8g15+x7g68+x6g101+x5g119\begin{array}{rcl}h&=&4h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&x_{1} g_{119}+x_{2} g_{101}+x_{3} g_{68}+x_{4} g_{15}\\ f&=&x_{8} g_{-15}+x_{7} g_{-68}+x_{6} g_{-101}+x_{5} g_{-119}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x5+x2x6+x3x7+x4x84)h8+(3x1x5+x2x6+x3x7+x4x86)h7+(4x1x5+2x2x6+2x3x78)h6+(5x1x5+3x2x6+2x3x710)h5+(6x1x5+4x2x6+2x3x712)h4+(4x1x5+3x2x6+x3x78)h3+(3x1x5+2x2x6+x3x76)h2+(2x1x5+2x2x64)h1[e,f] - h = \left(x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -4\right)h_{8}+\left(3x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -6\right)h_{7}+\left(4x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -8\right)h_{6}+\left(5x_{1} x_{5} +3x_{2} x_{6} +2x_{3} x_{7} -10\right)h_{5}+\left(6x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} -12\right)h_{4}+\left(4x_{1} x_{5} +3x_{2} x_{6} +x_{3} x_{7} -8\right)h_{3}+\left(3x_{1} x_{5} +2x_{2} x_{6} +x_{3} x_{7} -6\right)h_{2}+\left(2x_{1} x_{5} +2x_{2} x_{6} -4\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
2x1x5+2x2x64=03x1x5+2x2x6+x3x76=04x1x5+3x2x6+x3x78=06x1x5+4x2x6+2x3x712=05x1x5+3x2x6+2x3x710=04x1x5+2x2x6+2x3x78=03x1x5+x2x6+x3x7+x4x86=0x1x5+x2x6+x3x7+x4x84=0\begin{array}{rcl}2x_{1} x_{5} +2x_{2} x_{6} -4&=&0\\3x_{1} x_{5} +2x_{2} x_{6} +x_{3} x_{7} -6&=&0\\4x_{1} x_{5} +3x_{2} x_{6} +x_{3} x_{7} -8&=&0\\6x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} -12&=&0\\5x_{1} x_{5} +3x_{2} x_{6} +2x_{3} x_{7} -10&=&0\\4x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -8&=&0\\3x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -6&=&0\\x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -4&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=4h8+6h7+8h6+10h5+12h4+8h3+6h2+4h1e=x1g119+x2g101+x3g68+x4g15f=g15+g68+g101+g119\begin{array}{rcl}h&=&4h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\\e&=&x_{1} g_{119}+x_{2} g_{101}+x_{3} g_{68}+x_{4} g_{15}\\f&=&g_{-15}+g_{-68}+g_{-101}+g_{-119}\end{array}
Matrix form of the system we are trying to solve: (22003210431064205320422031111111)[col. vect.]=(4681210864)\begin{pmatrix}2 & 2 & 0 & 0\\ 3 & 2 & 1 & 0\\ 4 & 3 & 1 & 0\\ 6 & 4 & 2 & 0\\ 5 & 3 & 2 & 0\\ 4 & 2 & 2 & 0\\ 3 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}4\\ 6\\ 8\\ 12\\ 10\\ 8\\ 6\\ 4\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=4h8+6h7+8h6+10h5+12h4+8h3+6h2+4h1e=x1g119+x2g101+x3g68+x4g15f=x8g15+x7g68+x6g101+x5g119\begin{array}{rcl}h&=&4h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&x_{1} g_{119}+x_{2} g_{101}+x_{3} g_{68}+x_{4} g_{15}\\ f&=&x_{8} g_{-15}+x_{7} g_{-68}+x_{6} g_{-101}+x_{5} g_{-119}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
2x1x5+2x2x64=03x1x5+2x2x6+x3x76=04x1x5+3x2x6+x3x78=06x1x5+4x2x6+2x3x712=05x1x5+3x2x6+2x3x710=04x1x5+2x2x6+2x3x78=03x1x5+x2x6+x3x7+x4x86=0x1x5+x2x6+x3x7+x4x84=0\begin{array}{rcl}2x_{1} x_{5} +2x_{2} x_{6} -4&=&0\\3x_{1} x_{5} +2x_{2} x_{6} +x_{3} x_{7} -6&=&0\\4x_{1} x_{5} +3x_{2} x_{6} +x_{3} x_{7} -8&=&0\\6x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} -12&=&0\\5x_{1} x_{5} +3x_{2} x_{6} +2x_{3} x_{7} -10&=&0\\4x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -8&=&0\\3x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -6&=&0\\x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -4&=&0\\\end{array}

A14A^{4}_1
h-characteristic: (0, 1, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: 4A14A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 8V3ψ+28V2ψ+48Vψ+36V08V_{3\psi}+28V_{2\psi}+48V_{\psi}+36V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=3h8+6h7+9h6+12h5+15h4+10h3+8h2+5h1e=g114+g106+g91+g69f=g69+g91+g106+g114\begin{array}{rcl}h&=&3h_{8}+6h_{7}+9h_{6}+12h_{5}+15h_{4}+10h_{3}+8h_{2}+5h_{1}\\ e&=&g_{114}+g_{106}+g_{91}+g_{69}\\ f&=&g_{-69}+g_{-91}+g_{-106}+g_{-114}\end{array}
Lie brackets of the above elements.
h=3h8+6h7+9h6+12h5+15h4+10h3+8h2+5h1e=g114+g106+g91+g69f=g69+g91+g106+g114\begin{array}{rcl}[e, f]&=&3h_{8}+6h_{7}+9h_{6}+12h_{5}+15h_{4}+10h_{3}+8h_{2}+5h_{1}\\ [h, e]&=&2g_{114}+2g_{106}+2g_{91}+2g_{69}\\ [h, f]&=&-2g_{-69}-2g_{-91}-2g_{-106}-2g_{-114}\end{array}
Centralizer type: C4C_4
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 36): g8g_{-8}, g6g_{-6}, g4g_{-4}, g1g_{-1}, h1h_{1}, h4h_{4}, h6h_{6}, h8h_{8}, g1g_{1}, g3+g16g_{3}+g_{-16}, g4g_{4}, g5+g20g_{5}+g_{-20}, g6g_{6}, g7+g22g_{7}+g_{-22}, g8g_{8}, g9g11g_{9}-g_{-11}, g11g9g_{11}-g_{-9}, g12g13g_{12}-g_{-13}, g13g12g_{13}-g_{-12}, g14g15g_{14}-g_{-15}, g15g14g_{15}-g_{-14}, g16+g3g_{16}+g_{-3}, g19+g32g_{19}+g_{-32}, g20+g5g_{20}+g_{-5}, g21+g36g_{21}+g_{-36}, g22+g7g_{22}+g_{-7}, g24g27g_{24}-g_{-27}, g27g24g_{27}-g_{-24}, g28g29g_{28}-g_{-29}, g29g28g_{29}-g_{-28}, g32+g19g_{32}+g_{-19}, g35+g49g_{35}+g_{-49}, g36+g21g_{36}+g_{-21}, g40g43g_{40}-g_{-43}, g43g40g_{43}-g_{-40}, g49+g35g_{49}+g_{-35}
Basis of centralizer intersected with cartan (dimension: 4): h8-h_{8}, h6-h_{6}, h4-h_{4}, h1-h_{1}
Cartan of centralizer (dimension: 4): h8-h_{8}, h6-h_{6}, h4-h_{4}, h1-h_{1}
Cartan-generating semisimple element: h811h6+9h4+7h1-h_{8}-11h_{6}+9h_{4}+7h_{1}
adjoint action: (200000000000000000000000000000000000022000000000000000000000000000000000000180000000000000000000000000000000000001400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001400000000000000000000000000000000000016000000000000000000000000000000000000180000000000000000000000000000000000002000000000000000000000000000000000000220000000000000000000000000000000000001200000000000000000000000000000000000020000000000000000000000000000000000002000000000000000000000000000000000000200000000000000000000000000000000000020000000000000000000000000000000000000200000000000000000000000000000000000001000000000000000000000000000000000000010000000000000000000000000000000000000160000000000000000000000000000000000004000000000000000000000000000000000000200000000000000000000000000000000000080000000000000000000000000000000000001200000000000000000000000000000000000018000000000000000000000000000000000000180000000000000000000000000000000000001000000000000000000000000000000000000010000000000000000000000000000000000000400000000000000000000000000000000000060000000000000000000000000000000000008000000000000000000000000000000000000800000000000000000000000000000000000080000000000000000000000000000000000006)\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 22 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -22 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 20 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -20 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6\\ \end{pmatrix}
Characteristic polynomial ad H: x362520x34+2795856x321802978560x30+751764496896x28213405046493184x26+42325204854644736x245927374146759819264x22+585735112548154146816x2040395580235105269448704x18+1900914578292821002813440x1658802679826719891095814144x14+1129661707177787121970708480x1212448755958900651791822618624x10+72076614805820800074409574400x8203592982265266914365276160000x6+221725839307140545642496000000x4x^{36}-2520x^{34}+2795856x^{32}-1802978560x^{30}+751764496896x^{28}-213405046493184x^{26}+42325204854644736x^{24}-5927374146759819264x^{22}+585735112548154146816x^{20}-40395580235105269448704x^{18}+1900914578292821002813440x^{16}-58802679826719891095814144x^{14}+1129661707177787121970708480x^{12}-12448755958900651791822618624x^{10}+72076614805820800074409574400x^8-203592982265266914365276160000x^6+221725839307140545642496000000x^4
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -22)(x -20)(x -18)(x -18)(x -16)(x -14)(x -12)(x -10)(x -10)(x -8)(x -8)(x -6)(x -4)(x -2)(x -2)(x -2)(x +2)(x +2)(x +2)(x +4)(x +6)(x +8)(x +8)(x +10)(x +10)(x +12)(x +14)(x +16)(x +18)(x +18)(x +20)(x +22)
Eigenvalues of ad H: 00, 2222, 2020, 1818, 1616, 1414, 1212, 1010, 88, 66, 44, 22, 2-2, 4-4, 6-6, 8-8, 10-10, 12-12, 14-14, 16-16, 18-18, 20-20, 22-22
36 eigenvectors of ad H: 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
Centralizer type: C^{1}_4
Reductive components (1 total):
Scalar product computed: (1601120160011201600112016001300011200160)\begin{pmatrix}1/60 & -1/120 & -1/60 & 0\\ -1/120 & 1/60 & 0 & -1/120\\ -1/60 & 0 & 1/30 & 0\\ 0 & -1/120 & 0 & 1/60\\ \end{pmatrix}
Simple basis of Cartan of centralizer (4 total):
h8+h1h_{8}+h_{1}
matching e: g49+g35g_{49}+g_{-35}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000002000000000000000000000000000000000000100000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000001000000000000000000000000000000000000100000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000010000000000000000000000000000000000001000000000000000000000000000000000000100000000000000000000000000000000000010000000000000000000000000000000000001000000000000000000000000000000000000100000000000000000000000000000000000020000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002)\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}
h4h1h_{4}-h_{1}
matching e: g11g9g_{11}-g_{-9}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000200000000000000000000000000000000000020000000000000000000000000000000000001000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000001000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000100000000000000000000000000000000000010000000000000000000000000000000000001000000000000000000000000000000000000100000000000000000000000000000000000010000000000000000000000000000000000001000000000000000000000000000000000000100000000000000000000000000000000000010000000000000000000000000000000000001)\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}
h8-h_{8}
matching e: g8g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: (200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000001000000000000000000000000000000000000100000000000000000000000000000000000010000000000000000000000000000000000001)\begin{pmatrix}2 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h6h4-h_{6}-h_{4}
matching e: g5+g20g_{5}+g_{-20}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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Linear space basis of intersection of centralizer and ambient Cartan:
h8+h1h_{8}+h_{1}
matching e: g49+g35g_{49}+g_{-35}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h4h1h_{4}-h_{1}
matching e: g11g9g_{11}-g_{-9}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h8-h_{8}
matching e: g8g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h6h4-h_{6}-h_{4}
matching e: g5+g20g_{5}+g_{-20}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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Elements in Cartan dual to root system: (1, 1, 1, 1), (-1, -1, -1, -1), (2, 2, 2, 1), (-2, -2, -2, -1), (2, 1, 2, 1), (-2, -1, -2, -1), (1, 1, 1, 0), (-1, -1, -1, 0), (2, 1, 2, 0), (-2, -1, -2, 0), (1, 0, 1, 0), (-1, 0, -1, 0), (1, 1, 2, 1), (-1, -1, -2, -1), (1, 1, 0, 1), (-1, -1, 0, -1), (1, 1, 2, 0), (-1, -1, -2, 0), (1, 1, 0, 0), (-1, -1, 0, 0), (1, 0, 2, 0), (-1, 0, -2, 0), (1, 0, 0, 0), (-1, 0, 0, 0), (0, 1, 0, 1), (0, -1, 0, -1), (0, 0, 1, 0), (0, 0, -1, 0), (0, 0, 0, 1), (0, 0, 0, -1), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (240120120012024001201200120001200240)\begin{pmatrix}240 & -120 & -120 & 0\\ -120 & 240 & 0 & -120\\ -120 & 0 & 120 & 0\\ 0 & -120 & 0 & 240\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=3h8+6h7+9h6+12h5+15h4+10h3+8h2+5h1e=x1g114+x2g106+x3g91+x4g69e=x8g69+x7g91+x6g106+x5g114\begin{array}{rcl}h&=&3h_{8}+6h_{7}+9h_{6}+12h_{5}+15h_{4}+10h_{3}+8h_{2}+5h_{1}\\ e&=&x_{1} g_{114}+x_{2} g_{106}+x_{3} g_{91}+x_{4} g_{69}\\ f&=&x_{8} g_{-69}+x_{7} g_{-91}+x_{6} g_{-106}+x_{5} g_{-114}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x5+x2x6+x3x73)h8+(2x1x5+2x2x6+2x3x76)h7+(3x1x5+3x2x6+2x3x7+x4x89)h6+(4x1x5+4x2x6+2x3x7+2x4x812)h5+(5x1x5+4x2x6+3x3x7+3x4x815)h4+(4x1x5+2x2x6+2x3x7+2x4x810)h3+(2x1x5+2x2x6+2x3x7+2x4x88)h2+(2x1x5+x2x6+x3x7+x4x85)h1[e,f] - h = \left(x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} -3\right)h_{8}+\left(2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -6\right)h_{7}+\left(3x_{1} x_{5} +3x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -9\right)h_{6}+\left(4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12\right)h_{5}+\left(5x_{1} x_{5} +4x_{2} x_{6} +3x_{3} x_{7} +3x_{4} x_{8} -15\right)h_{4}+\left(4x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -10\right)h_{3}+\left(2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -8\right)h_{2}+\left(2x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -5\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
2x1x5+x2x6+x3x7+x4x85=02x1x5+2x2x6+2x3x7+2x4x88=04x1x5+2x2x6+2x3x7+2x4x810=05x1x5+4x2x6+3x3x7+3x4x815=04x1x5+4x2x6+2x3x7+2x4x812=03x1x5+3x2x6+2x3x7+x4x89=02x1x5+2x2x6+2x3x76=0x1x5+x2x6+x3x73=0\begin{array}{rcl}2x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -5&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -8&=&0\\4x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -10&=&0\\5x_{1} x_{5} +4x_{2} x_{6} +3x_{3} x_{7} +3x_{4} x_{8} -15&=&0\\4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\3x_{1} x_{5} +3x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -9&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -6&=&0\\x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} -3&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=3h8+6h7+9h6+12h5+15h4+10h3+8h2+5h1e=x1g114+x2g106+x3g91+x4g69f=g69+g91+g106+g114\begin{array}{rcl}h&=&3h_{8}+6h_{7}+9h_{6}+12h_{5}+15h_{4}+10h_{3}+8h_{2}+5h_{1}\\e&=&x_{1} g_{114}+x_{2} g_{106}+x_{3} g_{91}+x_{4} g_{69}\\f&=&g_{-69}+g_{-91}+g_{-106}+g_{-114}\end{array}
Matrix form of the system we are trying to solve: (21112222422254334422332122201110)[col. vect.]=(58101512963)\begin{pmatrix}2 & 1 & 1 & 1\\ 2 & 2 & 2 & 2\\ 4 & 2 & 2 & 2\\ 5 & 4 & 3 & 3\\ 4 & 4 & 2 & 2\\ 3 & 3 & 2 & 1\\ 2 & 2 & 2 & 0\\ 1 & 1 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}5\\ 8\\ 10\\ 15\\ 12\\ 9\\ 6\\ 3\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=3h8+6h7+9h6+12h5+15h4+10h3+8h2+5h1e=x1g114+x2g106+x3g91+x4g69f=x8g69+x7g91+x6g106+x5g114\begin{array}{rcl}h&=&3h_{8}+6h_{7}+9h_{6}+12h_{5}+15h_{4}+10h_{3}+8h_{2}+5h_{1}\\ e&=&x_{1} g_{114}+x_{2} g_{106}+x_{3} g_{91}+x_{4} g_{69}\\ f&=&x_{8} g_{-69}+x_{7} g_{-91}+x_{6} g_{-106}+x_{5} g_{-114}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
2x1x5+x2x6+x3x7+x4x85=02x1x5+2x2x6+2x3x7+2x4x88=04x1x5+2x2x6+2x3x7+2x4x810=05x1x5+4x2x6+3x3x7+3x4x815=04x1x5+4x2x6+2x3x7+2x4x812=03x1x5+3x2x6+2x3x7+x4x89=02x1x5+2x2x6+2x3x76=0x1x5+x2x6+x3x73=0\begin{array}{rcl}2x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -5&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -8&=&0\\4x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -10&=&0\\5x_{1} x_{5} +4x_{2} x_{6} +3x_{3} x_{7} +3x_{4} x_{8} -15&=&0\\4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\3x_{1} x_{5} +3x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -9&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -6&=&0\\x_{1} x_{5} +x_{2} x_{6} +x_{3} x_{7} -3&=&0\\\end{array}

A13A^{3}_1
h-characteristic: (0, 0, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 6
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: 3A13A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 2V3ψ+27V2ψ+52Vψ+55V02V_{3\psi}+27V_{2\psi}+52V_{\psi}+55V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=3h8+6h7+8h6+10h5+12h4+8h3+6h2+4h1e=g118+g104+g74f=g74+g104+g118\begin{array}{rcl}h&=&3h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&g_{118}+g_{104}+g_{74}\\ f&=&g_{-74}+g_{-104}+g_{-118}\end{array}
Lie brackets of the above elements.
h=3h8+6h7+8h6+10h5+12h4+8h3+6h2+4h1e=g118+g104+g74f=g74+g104+g118\begin{array}{rcl}[e, f]&=&3h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\\ [h, e]&=&2g_{118}+2g_{104}+2g_{74}\\ [h, f]&=&-2g_{-74}-2g_{-104}-2g_{-118}\end{array}
Centralizer type: F4+A1F_4+A_1
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 55): g31g_{-31}, g25g_{-25}, g19g_{-19}, g18g_{-18}, g17g_{-17}, g12g_{-12}, g11g_{-11}, g10g_{-10}, g8g_{-8}, g5g_{-5}, g4g_{-4}, g3g_{-3}, g2g_{-2}, h2h_{2}, h3h_{3}, h4h_{4}, h5h_{5}, h8h_{8}, g1+g44g_{1}+g_{-44}, g2g_{2}, g3g_{3}, g4g_{4}, g5g_{5}, g6g46g_{6}-g_{-46}, g8g_{8}, g9+g37g_{9}+g_{-37}, g10g_{10}, g11g_{11}, g12g_{12}, g13+g39g_{13}+g_{-39}, g16+g30g_{16}+g_{-30}, g17g_{17}, g18g_{18}, g19g_{19}, g20g33g_{20}-g_{-33}, g23g24g_{23}-g_{-24}, g24g23g_{24}-g_{-23}, g25g_{25}, g26+g27g_{26}+g_{-27}, g27+g26g_{27}+g_{-26}, g30+g16g_{30}+g_{-16}, g31g_{31}, g32g69g_{32}-g_{-69}, g33g20g_{33}-g_{-20}, g37+g9g_{37}+g_{-9}, g38+g63g_{38}+g_{-63}, g39+g13g_{39}+g_{-13}, g44+g1g_{44}+g_{-1}, g45g57g_{45}-g_{-57}, g46g6g_{46}-g_{-6}, g51+g52g_{51}+g_{-52}, g52+g51g_{52}+g_{-51}, g57g45g_{57}-g_{-45}, g63+g38g_{63}+g_{-38}, g69g32g_{69}-g_{-32}
Basis of centralizer intersected with cartan (dimension: 5): h4-h_{4}, h3-h_{3}, h2-h_{2}, h8-h_{8}, h5-h_{5}
Cartan of centralizer (dimension: 5): h2-h_{2}, h3-h_{3}, h4-h_{4}, h8-h_{8}, h5-h_{5}
Cartan-generating semisimple element: 7h84h5h411h3+9h27h_{8}-4h_{5}-h_{4}-11h_{3}+9h_{2}
adjoint action: (100000000000000000000000000000000000000000000000000000005000000000000000000000000000000000000000000000000000000024000000000000000000000000000000000000000000000000000000016000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000000000000000000001700000000000000000000000000000000000000000000000000000002300000000000000000000000000000000000000000000000000000001400000000000000000000000000000000000000000000000000000007000000000000000000000000000000000000000000000000000000040000000000000000000000000000000000000000000000000000000210000000000000000000000000000000000000000000000000000000190000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011000000000000000000000000000000000000000000000000000000019000000000000000000000000000000000000000000000000000000021000000000000000000000000000000000000000000000000000000040000000000000000000000000000000000000000000000000000000700000000000000000000000000000000000000000000000000000004000000000000000000000000000000000000000000000000000000014000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000023000000000000000000000000000000000000000000000000000000017000000000000000000000000000000000000000000000000000000030000000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000000000000000000006000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000000000160000000000000000000000000000000000000000000000000000000240000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000001300000000000000000000000000000000000000000000000000000001300000000000000000000000000000000000000000000000000000005000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000000000060000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000009000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000300000000000000000000000000000000000000000000000000000001100000000000000000000000000000000000000000000000000000001400000000000000000000000000000000000000000000000000000004000000000000000000000000000000000000000000000000000000070000000000000000000000000000000000000000000000000000000700000000000000000000000000000000000000000000000000000001400000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000009)\begin{pmatrix}1 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Characteristic polynomial ad H: x554030x53+7420707x518288960820x49+6289630704393x473441096630773586x45+1406418407803274739x43439116373135493827872x41+106237684081530119277675x3920091341534367769609640258x37+2984041478228684592335469089x35348576016954554529383558273444x33+31983077273280174824871489236763x312296225408801429296452193295152750x29+128192930441163178777428658725567177x275516547431402641511928460997724640584x25+180900744032538053837006411917211423856x234454733202945228552174971365049038926848x21+80857423544215564629250311450364780107520x191056027521397382748082542797288378396813312x17+9609045006153445911122675858403155632558080x1558209246747539850641323874448326792780906496x13+219317443122807651082952075578046618664960000x11461341900698606702265990023491645425582080000x9+455327460693180520512792463779928473600000000x7163723352055063895820634186663434649600000000x5x^{55}-4030x^{53}+7420707x^{51}-8288960820x^{49}+6289630704393x^{47}-3441096630773586x^{45}+1406418407803274739x^{43}-439116373135493827872x^{41}+106237684081530119277675x^{39}-20091341534367769609640258x^{37}+2984041478228684592335469089x^{35}-348576016954554529383558273444x^{33}+31983077273280174824871489236763x^{31}-2296225408801429296452193295152750x^{29}+128192930441163178777428658725567177x^{27}-5516547431402641511928460997724640584x^{25}+180900744032538053837006411917211423856x^{23}-4454733202945228552174971365049038926848x^{21}+80857423544215564629250311450364780107520x^{19}-1056027521397382748082542797288378396813312x^{17}+9609045006153445911122675858403155632558080x^{15}-58209246747539850641323874448326792780906496x^{13}+219317443122807651082952075578046618664960000x^{11}-461341900698606702265990023491645425582080000x^9+455327460693180520512792463779928473600000000x^7-163723352055063895820634186663434649600000000x^5
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x )(x -24)(x -23)(x -21)(x -20)(x -19)(x -17)(x -16)(x -14)(x -14)(x -13)(x -11)(x -10)(x -10)(x -9)(x -7)(x -7)(x -6)(x -5)(x -4)(x -4)(x -3)(x -3)(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(x +3)(x +3)(x +4)(x +4)(x +5)(x +6)(x +7)(x +7)(x +9)(x +10)(x +10)(x +11)(x +13)(x +14)(x +14)(x +16)(x +17)(x +19)(x +20)(x +21)(x +23)(x +24)
Eigenvalues of ad H: 00, 2424, 2323, 2121, 2020, 1919, 1717, 1616, 1414, 1313, 1111, 1010, 99, 77, 66, 55, 44, 33, 22, 11, 1-1, 2-2, 3-3, 4-4, 5-5, 6-6, 7-7, 9-9, 10-10, 11-11, 13-13, 14-14, 16-16, 17-17, 19-19, 20-20, 21-21, 23-23, 24-24
55 eigenvectors of ad H: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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Centralizer type: F^{1}_4+A^{1}_1
Reductive components (2 total):
Scalar product computed: (16000112001301600016013016011200160160)\begin{pmatrix}1/60 & 0 & 0 & -1/120\\ 0 & 1/30 & -1/60 & 0\\ 0 & -1/60 & 1/30 & -1/60\\ -1/120 & 0 & -1/60 & 1/60\\ \end{pmatrix}
Simple basis of Cartan of centralizer (4 total):
h5+h2h_{5}+h_{2}
matching e: g30+g16g_{30}+g_{-16}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h52h4h3h2-h_{5}-2h_{4}-h_{3}-h_{2}
matching e: g31g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
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h4+h3+h2h_{4}+h_{3}+h_{2}
matching e: g17g_{17}
verification: [h,e]2e=0[h,e]-2e=0
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h3h2-h_{3}-h_{2}
matching e: g20g33g_{20}-g_{-33}
verification: [h,e]2e=0[h,e]-2e=0
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Linear space basis of intersection of centralizer and ambient Cartan:
h5+h2h_{5}+h_{2}
matching e: g30+g16g_{30}+g_{-16}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h52h4h3h2-h_{5}-2h_{4}-h_{3}-h_{2}
matching e: g31g_{-31}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h4+h3+h2h_{4}+h_{3}+h_{2}
matching e: g17g_{17}
verification: [h,e]2e=0[h,e]-2e=0
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h3h2-h_{3}-h_{2}
matching e: g20g33g_{20}-g_{-33}
verification: [h,e]2e=0[h,e]-2e=0
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Elements in Cartan dual to root system: (1, 2, 3, 2), (-1, -2, -3, -2), (1, 1, 3, 2), (-1, -1, -3, -2), (1, 1, 2, 2), (-1, -1, -2, -2), (2, 2, 4, 3), (-2, -2, -4, -3), (1, 1, 2, 1), (-1, -1, -2, -1), (1, 1, 1, 1), (-1, -1, -1, -1), (1, 0, 1, 1), (-1, 0, -1, -1), (1, 2, 4, 3), (-1, -2, -4, -3), (1, 2, 4, 2), (-1, -2, -4, -2), (1, 2, 2, 2), (-1, -2, -2, -2), (1, 2, 2, 1), (-1, -2, -2, -1), (1, 0, 2, 2), (-1, 0, -2, -2), (1, 0, 2, 1), (-1, 0, -2, -1), (1, 0, 0, 1), (-1, 0, 0, -1), (0, 1, 2, 1), (0, -1, -2, -1), (1, 0, 0, 0), (-1, 0, 0, 0), (0, 1, 1, 1), (0, -1, -1, -1), (0, 0, 1, 1), (0, 0, -1, -1), (0, 2, 2, 1), (0, -2, -2, -1), (0, 1, 1, 0), (0, -1, -1, 0), (0, 0, 2, 1), (0, 0, -2, -1), (0, 0, 1, 0), (0, 0, -1, 0), (0, 0, 0, 1), (0, 0, 0, -1), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (2400012001206000601201201200120240)\begin{pmatrix}240 & 0 & 0 & -120\\ 0 & 120 & -60 & 0\\ 0 & -60 & 120 & -120\\ -120 & 0 & -120 & 240\\ \end{pmatrix}

Scalar product computed: (130)\begin{pmatrix}1/30\\ \end{pmatrix}
Simple basis of Cartan of centralizer (1 total):
h8h_{8}
matching e: g8g_{8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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Linear space basis of intersection of centralizer and ambient Cartan:
h8h_{8}
matching e: g8g_{8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120)\begin{pmatrix}120\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=3h8+6h7+8h6+10h5+12h4+8h3+6h2+4h1e=x1g118+x2g104+x3g74e=x6g74+x5g104+x4g118\begin{array}{rcl}h&=&3h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&x_{1} g_{118}+x_{2} g_{104}+x_{3} g_{74}\\ f&=&x_{6} g_{-74}+x_{5} g_{-104}+x_{4} g_{-118}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (x1x4+x2x5+x3x63)h8+(2x1x4+2x2x5+2x3x66)h7+(4x1x4+2x2x5+2x3x68)h6+(5x1x4+3x2x5+2x3x610)h5+(6x1x4+4x2x5+2x3x612)h4+(4x1x4+3x2x5+x3x68)h3+(3x1x4+2x2x5+x3x66)h2+(2x1x4+2x2x54)h1[e,f] - h = \left(x_{1} x_{4} +x_{2} x_{5} +x_{3} x_{6} -3\right)h_{8}+\left(2x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -6\right)h_{7}+\left(4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8\right)h_{6}+\left(5x_{1} x_{4} +3x_{2} x_{5} +2x_{3} x_{6} -10\right)h_{5}+\left(6x_{1} x_{4} +4x_{2} x_{5} +2x_{3} x_{6} -12\right)h_{4}+\left(4x_{1} x_{4} +3x_{2} x_{5} +x_{3} x_{6} -8\right)h_{3}+\left(3x_{1} x_{4} +2x_{2} x_{5} +x_{3} x_{6} -6\right)h_{2}+\left(2x_{1} x_{4} +2x_{2} x_{5} -4\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
2x1x4+2x2x54=03x1x4+2x2x5+x3x66=04x1x4+3x2x5+x3x68=06x1x4+4x2x5+2x3x612=05x1x4+3x2x5+2x3x610=04x1x4+2x2x5+2x3x68=02x1x4+2x2x5+2x3x66=0x1x4+x2x5+x3x63=0\begin{array}{rcl}2x_{1} x_{4} +2x_{2} x_{5} -4&=&0\\3x_{1} x_{4} +2x_{2} x_{5} +x_{3} x_{6} -6&=&0\\4x_{1} x_{4} +3x_{2} x_{5} +x_{3} x_{6} -8&=&0\\6x_{1} x_{4} +4x_{2} x_{5} +2x_{3} x_{6} -12&=&0\\5x_{1} x_{4} +3x_{2} x_{5} +2x_{3} x_{6} -10&=&0\\4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8&=&0\\2x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -6&=&0\\x_{1} x_{4} +x_{2} x_{5} +x_{3} x_{6} -3&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=3h8+6h7+8h6+10h5+12h4+8h3+6h2+4h1e=x1g118+x2g104+x3g74f=g74+g104+g118\begin{array}{rcl}h&=&3h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\\e&=&x_{1} g_{118}+x_{2} g_{104}+x_{3} g_{74}\\f&=&g_{-74}+g_{-104}+g_{-118}\end{array}
Matrix form of the system we are trying to solve: (220321431642532422222111)[col. vect.]=(4681210863)\begin{pmatrix}2 & 2 & 0\\ 3 & 2 & 1\\ 4 & 3 & 1\\ 6 & 4 & 2\\ 5 & 3 & 2\\ 4 & 2 & 2\\ 2 & 2 & 2\\ 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}4\\ 6\\ 8\\ 12\\ 10\\ 8\\ 6\\ 3\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=3h8+6h7+8h6+10h5+12h4+8h3+6h2+4h1e=x1g118+x2g104+x3g74f=x6g74+x5g104+x4g118\begin{array}{rcl}h&=&3h_{8}+6h_{7}+8h_{6}+10h_{5}+12h_{4}+8h_{3}+6h_{2}+4h_{1}\\ e&=&x_{1} g_{118}+x_{2} g_{104}+x_{3} g_{74}\\ f&=&x_{6} g_{-74}+x_{5} g_{-104}+x_{4} g_{-118}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
2x1x4+2x2x54=03x1x4+2x2x5+x3x66=04x1x4+3x2x5+x3x68=06x1x4+4x2x5+2x3x612=05x1x4+3x2x5+2x3x610=04x1x4+2x2x5+2x3x68=02x1x4+2x2x5+2x3x66=0x1x4+x2x5+x3x63=0\begin{array}{rcl}2x_{1} x_{4} +2x_{2} x_{5} -4&=&0\\3x_{1} x_{4} +2x_{2} x_{5} +x_{3} x_{6} -6&=&0\\4x_{1} x_{4} +3x_{2} x_{5} +x_{3} x_{6} -8&=&0\\6x_{1} x_{4} +4x_{2} x_{5} +2x_{3} x_{6} -12&=&0\\5x_{1} x_{4} +3x_{2} x_{5} +2x_{3} x_{6} -10&=&0\\4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8&=&0\\2x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -6&=&0\\x_{1} x_{4} +x_{2} x_{5} +x_{3} x_{6} -3&=&0\\\end{array}

A12A^{2}_1
h-characteristic: (1, 0, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 4
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: 2A12A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: 14V2ψ+64Vψ+78V014V_{2\psi}+64V_{\psi}+78V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=2h8+4h7+6h6+8h5+10h4+7h3+5h2+4h1e=g120+g97f=g97+g120\begin{array}{rcl}h&=&2h_{8}+4h_{7}+6h_{6}+8h_{5}+10h_{4}+7h_{3}+5h_{2}+4h_{1}\\ e&=&g_{120}+g_{97}\\ f&=&g_{-97}+g_{-120}\end{array}
Lie brackets of the above elements.
h=2h8+4h7+6h6+8h5+10h4+7h3+5h2+4h1e=g120+g97f=g97+g120\begin{array}{rcl}[e, f]&=&2h_{8}+4h_{7}+6h_{6}+8h_{5}+10h_{4}+7h_{3}+5h_{2}+4h_{1}\\ [h, e]&=&2g_{120}+2g_{97}\\ [h, f]&=&-2g_{-97}-2g_{-120}\end{array}
Centralizer type: B6B_6
Killing form square of Cartan element dual to ambient long root: 120
Basis of the centralizer (dimension: 78): g60g_{-60}, g54g_{-54}, g48g_{-48}, g46g_{-46}, g41g_{-41}, g39g_{-39}, g35g_{-35}, g34g_{-34}, g33g_{-33}, g31g_{-31}, g28g_{-28}, g27g_{-27}, g26g_{-26}, g25g_{-25}, g21g_{-21}, g20g_{-20}, g19g_{-19}, g18g_{-18}, g17g_{-17}, g14g_{-14}, g13g_{-13}, g12g_{-12}, g11g_{-11}, g10g_{-10}, g7g_{-7}, g6g_{-6}, g5g_{-5}, g4g_{-4}, g3g_{-3}, g2g_{-2}, h2h_{2}, h3h_{3}, h4h_{4}, h5h_{5}, h6h_{6}, h7h_{7}, g2g_{2}, g3g_{3}, g4g_{4}, g5g_{5}, g6g_{6}, g7g_{7}, g8+g74g_{8}+g_{-74}, g10g_{10}, g11g_{11}, g12g_{12}, g13g_{13}, g14g_{14}, g15g68g_{15}-g_{-68}, g17g_{17}, g18g_{18}, g19g_{19}, g20g_{20}, g21g_{21}, g22+g62g_{22}+g_{-62}, g25g_{25}, g26g_{26}, g27g_{27}, g28g_{28}, g29g56g_{29}-g_{-56}, g31g_{31}, g33g_{33}, g34g_{34}, g35g_{35}, g36+g50g_{36}+g_{-50}, g39g_{39}, g41g_{41}, g42g43g_{42}-g_{-43}, g43g42g_{43}-g_{-42}, g46g_{46}, g48g_{48}, g50+g36g_{50}+g_{-36}, g54g_{54}, g56g29g_{56}-g_{-29}, g60g_{60}, g62+g22g_{62}+g_{-22}, g68g15g_{68}-g_{-15}, g74+g8g_{74}+g_{-8}
Basis of centralizer intersected with cartan (dimension: 6): h7-h_{7}, h6-h_{6}, h5-h_{5}, h4-h_{4}, h3-h_{3}, h2-h_{2}
Cartan of centralizer (dimension: 6): g19g19-g_{19}-g_{-19}, h4-h_{4}, h3h2-h_{3}-h_{2}, h7-h_{7}, h5h2-h_{5}-h_{2}, h6+h2-h_{6}+h_{2}
Cartan-generating semisimple element: g19+2h7+25h6+17h5+13h4+9h3+h2+g19g_{19}+2h_{7}+25h_{6}+17h_{5}+13h_{4}+9h_{3}+h_{2}+g_{-19}
adjoint action: 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Characteristic polynomial ad H: x787568x76+26494390x7457072449492x72+84897531281269x7092792011132727396x68+77429683054880151688x6650607325908717822411920x64+26379521949670885621277362x6211111761646466878339827044464x60+3819636113600375849315318291908x581079444212243977198158904616707224x56+252203396328018055320074107206758530x5448921202961930425024874298416520208376x52+7902533750002433007528035124966735059272x501065289647531251470275448635411647833404368x48+119989178688523211131033330818345341580112909x4611297775851786451824318358345668921311317914720x44+888999501209511754542966994263273654069196916086x4258402064776707638515806273520002892769047346183732x40+3197430045443288504026136688857351460112281900634441x38145509182355946335066608383052167350403124244688016708x36+5484897795944149455048403524864418077505752707608502080x34170471728242409205772789523996453843695133057604396133632x32+4343362265511027675453617307416323791319662740433056771584x3090061553164229895089547235501073183408637165147689383540736x28+1506174736585143461493110222826761648494315753189419376918528x2620089803033532911055848149841278917447244254679546072195727360x24+210769755664435790860211467369682962431102314050956282498711552x221709401444891056524315534659080391320381556261507395977908322304x20+10485344664183787472368120258327801821724970387228360010262118400x1847293336021704133879927431020441049849185804648369408389939200000x16+151088914256162025652583606306310608931387070048189494067200000000x14324371018013087891721028643754579040243856000564559085568000000000x12+431933499517646926898590029091108755188435350125186908160000000000x10311108650650188508266167462593999538591221945357828096000000000000x8+90782547447831460786518553009532065958073005506560000000000000000x6x^{78}-7568x^{76}+26494390x^{74}-57072449492x^{72}+84897531281269x^{70}-92792011132727396x^{68}+77429683054880151688x^{66}-50607325908717822411920x^{64}+26379521949670885621277362x^{62}-11111761646466878339827044464x^{60}+3819636113600375849315318291908x^{58}-1079444212243977198158904616707224x^{56}+252203396328018055320074107206758530x^{54}-48921202961930425024874298416520208376x^{52}+7902533750002433007528035124966735059272x^{50}-1065289647531251470275448635411647833404368x^{48}+119989178688523211131033330818345341580112909x^{46}-11297775851786451824318358345668921311317914720x^{44}+888999501209511754542966994263273654069196916086x^{42}-58402064776707638515806273520002892769047346183732x^{40}+3197430045443288504026136688857351460112281900634441x^{38}-145509182355946335066608383052167350403124244688016708x^{36}+5484897795944149455048403524864418077505752707608502080x^{34}-170471728242409205772789523996453843695133057604396133632x^{32}+4343362265511027675453617307416323791319662740433056771584x^{30}-90061553164229895089547235501073183408637165147689383540736x^{28}+1506174736585143461493110222826761648494315753189419376918528x^{26}-20089803033532911055848149841278917447244254679546072195727360x^{24}+210769755664435790860211467369682962431102314050956282498711552x^{22}-1709401444891056524315534659080391320381556261507395977908322304x^{20}+10485344664183787472368120258327801821724970387228360010262118400x^{18}-47293336021704133879927431020441049849185804648369408389939200000x^{16}+151088914256162025652583606306310608931387070048189494067200000000x^{14}-324371018013087891721028643754579040243856000564559085568000000000x^{12}+431933499517646926898590029091108755188435350125186908160000000000x^{10}-311108650650188508266167462593999538591221945357828096000000000000x^8+90782547447831460786518553009532065958073005506560000000000000000x^6
Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x )(x )(x -32)(x -30)(x -27)(x -26)(x -25)(x -23)(x -21)(x -20)(x -19)(x -16)(x -16)(x -14)(x -13)(x -12)(x -11)(x -11)(x -10)(x -9)(x -9)(x -7)(x -7)(x -7)(x -6)(x -6)(x -5)(x -5)(x -5)(x -4)(x -4)(x -3)(x -3)(x -2)(x -2)(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(x +2)(x +2)(x +3)(x +3)(x +4)(x +4)(x +5)(x +5)(x +5)(x +6)(x +6)(x +7)(x +7)(x +7)(x +9)(x +9)(x +10)(x +11)(x +11)(x +12)(x +13)(x +14)(x +16)(x +16)(x +19)(x +20)(x +21)(x +23)(x +25)(x +26)(x +27)(x +30)(x +32)
Eigenvalues of ad H: 00, 3232, 3030, 2727, 2626, 2525, 2323, 2121, 2020, 1919, 1616, 1414, 1313, 1212, 1111, 1010, 99, 77, 66, 55, 44, 33, 22, 11, 1-1, 2-2, 3-3, 4-4, 5-5, 6-6, 7-7, 9-9, 10-10, 11-11, 12-12, 13-13, 14-14, 16-16, 19-19, 20-20, 21-21, 23-23, 25-25, 26-26, 27-27, 30-30, 32-32
78 eigenvectors of ad H: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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Centralizer type: B^{1}_6
Reductive components (1 total):
Scalar product computed: (13000016000130160160000160130016000160013001601600160013000001600160)\begin{pmatrix}1/30 & 0 & 0 & 0 & -1/60 & 0\\ 0 & 1/30 & -1/60 & -1/60 & 0 & 0\\ 0 & -1/60 & 1/30 & 0 & -1/60 & 0\\ 0 & -1/60 & 0 & 1/30 & 0 & -1/60\\ -1/60 & 0 & -1/60 & 0 & 1/30 & 0\\ 0 & 0 & 0 & -1/60 & 0 & 1/60\\ \end{pmatrix}
Simple basis of Cartan of centralizer (6 total):
12g19+h6+32h5+32h4+12h3+h212g19-1/2g_{19}+h_{6}+3/2h_{5}+3/2h_{4}+1/2h_{3}+h_{2}-1/2g_{-19}
matching e: g46g26g_{46}-g_{26}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h4-h_{4}
matching e: g4g_{-4}
verification: [h,e]2e=0[h,e]-2e=0
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12g1912h5+12h4+12h312g19-1/2g_{19}-1/2h_{5}+1/2h_{4}+1/2h_{3}-1/2g_{-19}
matching e: g11g5g_{11}-g_{-5}
verification: [h,e]2e=0[h,e]-2e=0
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g19+g19g_{19}+g_{-19}
matching e: g19h5h4h3g19g_{19}-h_{5}-h_{4}-h_{3}-g_{-19}
verification: [h,e]2e=0[h,e]-2e=0
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2h7+2h6+2h5+2h4+h3+h22h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}
matching e: g74+g8g_{74}+g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
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Linear space basis of intersection of centralizer and ambient Cartan:
12g19+h6+32h5+32h4+12h3+h212g19-1/2g_{19}+h_{6}+3/2h_{5}+3/2h_{4}+1/2h_{3}+h_{2}-1/2g_{-19}
matching e: g46g26g_{46}-g_{26}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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h4-h_{4}
matching e: g4g_{-4}
verification: [h,e]2e=0[h,e]-2e=0
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12g1912h5+12h4+12h312g19-1/2g_{19}-1/2h_{5}+1/2h_{4}+1/2h_{3}-1/2g_{-19}
matching e: g11g5g_{11}-g_{-5}
verification: [h,e]2e=0[h,e]-2e=0
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h7h6h5h4h3h2-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}
matching e: g41g_{-41}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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g19+g19g_{19}+g_{-19}
matching e: g19h5h4h3g19g_{19}-h_{5}-h_{4}-h_{3}-g_{-19}
verification: [h,e]2e=0[h,e]-2e=0
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2h7+2h6+2h5+2h4+h3+h22h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}
matching e: g74+g8g_{74}+g_{-8}
verification: [h,e]2e=0[h,e]-2e=0
adjoint action: 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Elements in Cartan dual to root system: (1, 2, 2, 2, 2, 1), (-1, -2, -2, -2, -2, -1), (1, 2, 2, 2, 1, 1), (-1, -2, -2, -2, -1, -1), (1, 2, 1, 2, 1, 1), (-1, -2, -1, -2, -1, -1), (1, 1, 1, 2, 1, 1), (-1, -1, -1, -2, -1, -1), (1, 1, 1, 1, 1, 1), (-1, -1, -1, -1, -1, -1), (2, 2, 2, 2, 2, 1), (-2, -2, -2, -2, -2, -1), (1, 1, 1, 1, 1, 0), (-1, -1, -1, -1, -1, 0), (1, 1, 1, 0, 1, 0), (-1, -1, -1, 0, -1, 0), (1, 0, 1, 0, 1, 0), (-1, 0, -1, 0, -1, 0), (1, 0, 0, 0, 1, 0), (-1, 0, 0, 0, -1, 0), (0, 2, 2, 2, 1, 1), (0, -2, -2, -2, -1, -1), (1, 0, 0, 0, 0, 0), (-1, 0, 0, 0, 0, 0), (0, 2, 1, 2, 1, 1), (0, -2, -1, -2, -1, -1), (0, 1, 1, 2, 1, 1), (0, -1, -1, -2, -1, -1), (0, 1, 1, 1, 1, 1), (0, -1, -1, -1, -1, -1), (0, 2, 1, 2, 0, 1), (0, -2, -1, -2, 0, -1), (0, 1, 1, 2, 0, 1), (0, -1, -1, -2, 0, -1), (0, 2, 2, 2, 2, 1), (0, -2, -2, -2, -2, -1), (0, 1, 1, 1, 0, 1), (0, -1, -1, -1, 0, -1), (0, 1, 1, 1, 1, 0), (0, -1, -1, -1, -1, 0), (0, 2, 2, 2, 0, 1), (0, -2, -2, -2, 0, -1), (0, 1, 0, 2, 0, 1), (0, -1, 0, -2, 0, -1), (0, 1, 1, 0, 1, 0), (0, -1, -1, 0, -1, 0), (0, 1, 0, 1, 0, 1), (0, -1, 0, -1, 0, -1), (0, 0, 1, 0, 1, 0), (0, 0, -1, 0, -1, 0), (0, 1, 1, 1, 0, 0), (0, -1, -1, -1, 0, 0), (0, 0, 0, 1, 0, 1), (0, 0, 0, -1, 0, -1), (0, 1, 1, 0, 0, 0), (0, -1, -1, 0, 0, 0), (0, 2, 0, 2, 0, 1), (0, -2, 0, -2, 0, -1), (0, 0, 1, 0, 0, 0), (0, 0, -1, 0, 0, 0), (0, 0, 0, 2, 0, 1), (0, 0, 0, -2, 0, -1), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, -1, 0), (0, 1, 0, 1, 0, 0), (0, -1, 0, -1, 0, 0), (0, 0, 0, 0, 0, 1), (0, 0, 0, 0, 0, -1), (0, 0, 0, 1, 0, 0), (0, 0, 0, -1, 0, 0), (0, 1, 0, 0, 0, 0), (0, -1, 0, 0, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form: (120000600012060600006012006000600120012060060012000001200240)\begin{pmatrix}120 & 0 & 0 & 0 & -60 & 0\\ 0 & 120 & -60 & -60 & 0 & 0\\ 0 & -60 & 120 & 0 & -60 & 0\\ 0 & -60 & 0 & 120 & 0 & -120\\ -60 & 0 & -60 & 0 & 120 & 0\\ 0 & 0 & 0 & -120 & 0 & 240\\ \end{pmatrix}
Unfold the hidden panel for more information.

Unknown elements.
h=2h8+4h7+6h6+8h5+10h4+7h3+5h2+4h1e=x1g120+x2g97e=x4g97+x3g120\begin{array}{rcl}h&=&2h_{8}+4h_{7}+6h_{6}+8h_{5}+10h_{4}+7h_{3}+5h_{2}+4h_{1}\\ e&=&x_{1} g_{120}+x_{2} g_{97}\\ f&=&x_{4} g_{-97}+x_{3} g_{-120}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (2x1x32)h8+(3x1x3+x2x44)h7+(4x1x3+2x2x46)h6+(5x1x3+3x2x48)h5+(6x1x3+4x2x410)h4+(4x1x3+3x2x47)h3+(3x1x3+2x2x45)h2+(2x1x3+2x2x44)h1[e,f] - h = \left(2x_{1} x_{3} -2\right)h_{8}+\left(3x_{1} x_{3} +x_{2} x_{4} -4\right)h_{7}+\left(4x_{1} x_{3} +2x_{2} x_{4} -6\right)h_{6}+\left(5x_{1} x_{3} +3x_{2} x_{4} -8\right)h_{5}+\left(6x_{1} x_{3} +4x_{2} x_{4} -10\right)h_{4}+\left(4x_{1} x_{3} +3x_{2} x_{4} -7\right)h_{3}+\left(3x_{1} x_{3} +2x_{2} x_{4} -5\right)h_{2}+\left(2x_{1} x_{3} +2x_{2} x_{4} -4\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
2x1x3+2x2x44=03x1x3+2x2x45=04x1x3+3x2x47=06x1x3+4x2x410=05x1x3+3x2x48=04x1x3+2x2x46=03x1x3+x2x44=02x1x32=0\begin{array}{rcl}2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\3x_{1} x_{3} +2x_{2} x_{4} -5&=&0\\4x_{1} x_{3} +3x_{2} x_{4} -7&=&0\\6x_{1} x_{3} +4x_{2} x_{4} -10&=&0\\5x_{1} x_{3} +3x_{2} x_{4} -8&=&0\\4x_{1} x_{3} +2x_{2} x_{4} -6&=&0\\3x_{1} x_{3} +x_{2} x_{4} -4&=&0\\2x_{1} x_{3} -2&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=2h8+4h7+6h6+8h5+10h4+7h3+5h2+4h1e=x1g120+x2g97f=g97+g120\begin{array}{rcl}h&=&2h_{8}+4h_{7}+6h_{6}+8h_{5}+10h_{4}+7h_{3}+5h_{2}+4h_{1}\\e&=&x_{1} g_{120}+x_{2} g_{97}\\f&=&g_{-97}+g_{-120}\end{array}
Matrix form of the system we are trying to solve: (2232436453423120)[col. vect.]=(457108642)\begin{pmatrix}2 & 2\\ 3 & 2\\ 4 & 3\\ 6 & 4\\ 5 & 3\\ 4 & 2\\ 3 & 1\\ 2 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}4\\ 5\\ 7\\ 10\\ 8\\ 6\\ 4\\ 2\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=2h8+4h7+6h6+8h5+10h4+7h3+5h2+4h1e=x1g120+x2g97f=x4g97+x3g120\begin{array}{rcl}h&=&2h_{8}+4h_{7}+6h_{6}+8h_{5}+10h_{4}+7h_{3}+5h_{2}+4h_{1}\\ e&=&x_{1} g_{120}+x_{2} g_{97}\\ f&=&x_{4} g_{-97}+x_{3} g_{-120}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
2x1x3+2x2x44=03x1x3+2x2x45=04x1x3+3x2x47=06x1x3+4x2x410=05x1x3+3x2x48=04x1x3+2x2x46=03x1x3+x2x44=02x1x32=0\begin{array}{rcl}2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\3x_{1} x_{3} +2x_{2} x_{4} -5&=&0\\4x_{1} x_{3} +3x_{2} x_{4} -7&=&0\\6x_{1} x_{3} +4x_{2} x_{4} -10&=&0\\5x_{1} x_{3} +3x_{2} x_{4} -8&=&0\\4x_{1} x_{3} +2x_{2} x_{4} -6&=&0\\3x_{1} x_{3} +x_{2} x_{4} -4&=&0\\2x_{1} x_{3} -2&=&0\\\end{array}

A1A_1
h-characteristic: (0, 0, 0, 0, 0, 0, 0, 1)
Length of the weight dual to h: 2
Simple basis ambient algebra w.r.t defining h: 8 vectors: (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1: A1A^{1}_1
sl(2)sl{}\left(2\right)-module decomposition of the ambient Lie algebra: V2ψ+56Vψ+133V0V_{2\psi}+56V_{\psi}+133V_{0}
Below is one possible realization of the sl(2) subalgebra.
h=2h8+3h7+4h6+5h5+6h4+4h3+3h2+2h1e=g120f=g120\begin{array}{rcl}h&=&2h_{8}+3h_{7}+4h_{6}+5h_{5}+6h_{4}+4h_{3}+3h_{2}+2h_{1}\\ e&=&g_{120}\\ f&=&g_{-120}\end{array}
Lie brackets of the above elements.
h=2h8+3h7+4h6+5h5+6h4+4h3+3h2+2h1e=g120f=g120\begin{array}{rcl}[e, f]&=&2h_{8}+3h_{7}+4h_{6}+5h_{5}+6h_{4}+4h_{3}+3h_{2}+2h_{1}\\ [h, e]&=&2g_{120}\\ [h, f]&=&-2g_{-120}\end{array}
Centralizer type: E7E_7
Unfold the hidden panel for more information.

Unknown elements.
h=2h8+3h7+4h6+5h5+6h4+4h3+3h2+2h1e=x1g120e=x2g120\begin{array}{rcl}h&=&2h_{8}+3h_{7}+4h_{6}+5h_{5}+6h_{4}+4h_{3}+3h_{2}+2h_{1}\\ e&=&x_{1} g_{120}\\ f&=&x_{2} g_{-120}\end{array}
Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.
[e,f]h= (2x1x22)h8+(3x1x23)h7+(4x1x24)h6+(5x1x25)h5+(6x1x26)h4+(4x1x24)h3+(3x1x23)h2+(2x1x22)h1[e,f] - h = \left(2x_{1} x_{2} -2\right)h_{8}+\left(3x_{1} x_{2} -3\right)h_{7}+\left(4x_{1} x_{2} -4\right)h_{6}+\left(5x_{1} x_{2} -5\right)h_{5}+\left(6x_{1} x_{2} -6\right)h_{4}+\left(4x_{1} x_{2} -4\right)h_{3}+\left(3x_{1} x_{2} -3\right)h_{2}+\left(2x_{1} x_{2} -2\right)h_{1}
The polynomial system that corresponds to finding the h, e, f triple:
2x1x22=03x1x23=04x1x24=06x1x26=05x1x25=04x1x24=03x1x23=02x1x22=0\begin{array}{rcl}2x_{1} x_{2} -2&=&0\\3x_{1} x_{2} -3&=&0\\4x_{1} x_{2} -4&=&0\\6x_{1} x_{2} -6&=&0\\5x_{1} x_{2} -5&=&0\\4x_{1} x_{2} -4&=&0\\3x_{1} x_{2} -3&=&0\\2x_{1} x_{2} -2&=&0\\\end{array}
Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.
h=2h8+3h7+4h6+5h5+6h4+4h3+3h2+2h1e=x1g120f=g120\begin{array}{rcl}h&=&2h_{8}+3h_{7}+4h_{6}+5h_{5}+6h_{4}+4h_{3}+3h_{2}+2h_{1}\\e&=&x_{1} g_{120}\\f&=&g_{-120}\end{array}
Matrix form of the system we are trying to solve: (23465432)[col. vect.]=(23465432)\begin{pmatrix}2\\ 3\\ 4\\ 6\\ 5\\ 4\\ 3\\ 2\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}2\\ 3\\ 4\\ 6\\ 5\\ 4\\ 3\\ 2\\ \end{pmatrix}
The unknown Kostant-Sekiguchi elements.
h=2h8+3h7+4h6+5h5+6h4+4h3+3h2+2h1e=x1g120f=x2g120\begin{array}{rcl}h&=&2h_{8}+3h_{7}+4h_{6}+5h_{5}+6h_{4}+4h_{3}+3h_{2}+2h_{1}\\ e&=&x_{1} g_{120}\\ f&=&x_{2} g_{-120}\end{array}
ef=0e-f=0
θ(ef)=0\theta(e-f)=0
The polynomial system we need to solve.
2x1x22=03x1x23=04x1x24=06x1x26=05x1x25=04x1x24=03x1x23=02x1x22=0\begin{array}{rcl}2x_{1} x_{2} -2&=&0\\3x_{1} x_{2} -3&=&0\\4x_{1} x_{2} -4&=&0\\6x_{1} x_{2} -6&=&0\\5x_{1} x_{2} -5&=&0\\4x_{1} x_{2} -4&=&0\\3x_{1} x_{2} -3&=&0\\2x_{1} x_{2} -2&=&0\\\end{array}